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\centerline{\bf MULTIPLE WIENER--IT\^O INTEGRALS}
\centerline{with applications to limit theorems}
\smallskip
\centerline{\it P\'eter Major}

\bigskip\noindent
\null\hskip1cm {\it Lecture Notes in Mathematics}\/ \ 849

\noindent
\null\hskip1cm Springer--Verlag, Berlin, Heidelberg, New York (1981)

\bigskip\bigskip\bigskip\bigskip

\centerline {\script TABLE OF CONTENTS}
$$
\vbox{\halign{\hfill # \ &\vtop{\hsize=12truecm\parindent=0pt #
\vskip3pt} \quad &\vtop{\hsize=0.5truecm\parindent=0pt #
\vskip3pt} \cr
   & Introduction. \dotfill &\rightline{ii}\cr
   \noalign{\medskip}
1. & On a limit problem. \dotfill & \rightline{1}\cr
   & {\it Attachment to Section~1.} A brief overview about
     some results on generalized functions.
     \dotfill &  \vskip5pt \rightline{5}\cr
     \noalign{\smallskip}
2. & Wick polynomials. \dotfill & \rightline{8}\cr
3. & Random spectral measures. \dotfill & \rightline{13}\cr
   & {\it Attachment to Section~3.} A more detailed discussion about
     the spectral representation of the covariance function of stationary
     random fields. \dotfill &  \vskip16pt \rightline{18}\cr
     \noalign{\smallskip}
4. & Multiple Wiener--It\^o integrals. \dotfill & \rightline{23}\cr
5. & The proof of It\^o's formula. The diagram formula and some of
     its consequences. \dotfill & \vskip5pt \rightline{36} \cr
6. & Subordinated random fields. Construction of self-similar fields.
     \dotfill & \rightline{52}\cr
7. & On the original Wiener--It\^o integral. \dotfill & \rightline{64}\cr
8. & Non-central limit theorems. \dotfill & \rightline{69}\cr
9. &  History of the problem. Comments. \dotfill & \rightline{87}\cr
   \noalign{\medskip}
   & References. \dotfill & \rightline{95}\cr}}
$$

\vfill\eject

\beginsection Introduction.

One of the most important problems in probability theory is the
investigation of the limit distribution of partial sums of
appropriately normalized random variables. The case where the random
variables are independent is fairly well understood. Many results are
known also in the case where independence is replaced by an appropriate
mixing condition or some other ``almost independence'' property. Much
less is known about the limit behaviour of partial sums of really
dependent random variables. On the other hand, this case is becoming
more and more important, not only in probability theory, but also in
some applications in statistical physics.

The problem about the asymptotic behaviour of partial sums of dependent
random variables leads to the investigation of some very complicated
transformations of probability measures. The classical methods of
probability theory do not seem to work for this problem. On the other
hand, although we are still very far from a satisfactory solution
of this problem, we can already present some nontrivial results.

The so-called multiple Wiener--It\^o integrals have proved to be a
very useful tool in the investigation of this problem. The proofs of
almost all rigorous results in this field are closely related to this
technique. The notion of multiple Wiener--It\^o integrals was worked
out for the investigation of non-linear functionals over Gaussian fields.
It is closely related to the so-called Wick polynomials which can be
considered as the multi-dimensional generalization of Hermite polynomials.
The notion of Wick polynomials and multiple Wiener--It\^o integrals
were worked out at the same time and independently of each other.
Actually,  we discuss a modified version of the multiple Wiener--It\^o
integrals in greatest detail. The technical changes needed in the
definition of these modified integrals are not essential. On the other
hand, these modified integrals are more appropriate for certain
investigations, since they enable us to describe the action of shift
transformations and to apply some sort of random Fourier analysis.
There is also some connection between multiple Wiener--It\^o integrals
and the classical stochastic It\^o integrals. The main difference
between them is that in the first case deterministic functions are
integrated, and in the second case so-called non-anticipating functionals.
The consequence of this difference is that no technical  difficulty
arises when we want to define multiple Wiener--It\^o integrals  in the
multi-dimensional time case. On the other hand, a large class of
nonlinear functionals over Gaussian fields can be represented by means
$I_{n,k}(f)$ of multiple Wiener--It\^o integrals.

In this work we are interested in limit problems for sums of dependent
random variables. It is useful to consider this problem together with
its continuous time version. The natural formulation of the continuous
time version of this problem can be given by means of generalized fields.
Consequently we also have to discuss some questions about generalized
fields.

I have not tried to formulate all the results in the most general form.
My main goal was to work out the most important techniques needed in
the investigation of such problems. This is the reason why the greatest
part of this work deals with multiple Wiener--It\^o integrals. I have
tried to give a self-contained exposition of this subject and also
to explain the motivation behind the results.

I had the opportunity to participate in the Dobrushin--Sinai seminar
in Moscow. What I learned there was very useful also for the
preparation of this Lecture Note. Therefore I would like to thank
the members of this seminar for what I could learn from them, especially
P.~M.~Bleher, R.~L.~Dobrushin and Ya.~G.~Sinai.

\medskip\noindent
{\it Some remarks to this text.}

\medskip\noindent
This text is a slightly modified version of my Lecture Note {\it
Multiple Wiener--It\^o integrals with applications to limit theorems}\/
published in the {\it Lecture Notes in Mathematics}\/ series
(number~849) of the Springer Verlag in~1981. I decided to make a
special lecture on the basis of this work in the first semester of
the university course in~2011--2012 at the University of~Szeged.
Preparing for it I observed how difficult the reading of formulas
in this Lecture Note is. These difficulties arose because this
Lecture Note was written at the time when the $\TeX$ program still
did not exist, and the highest technical level of typing was
writing on an IBM machine that enabled one to type beside the usual
text also mathematical formulas. But the texts written in such a
way are very hard to read. To make my text more readable I decided
to retype it by means of the $\TeX$ program. This demanded some
changes. It implied e.g.\ to follow such partly typographical
partly linguistic rules by which one does not start a sentence
with a formula. Beside this, it suggested to formulate the basic
definitions in a (typographically) more explicit form and not as an
explanation inside the text. When typing this work I also tried to
rethink what I had written, to correct the errors and to make the
proofs more understandable. It was surprising and a little bit
shocking to meet my old personality by studying my old Lecture
Note and to recognize how much I have changed. Now I would expose
many details in a different way. Naturally I would also make many
changes by taking into account the results proved since the time I
wrote this note. Nevertheless I decided to make no essential
changes in the text, to restrict myself to the correction of the
errors I found, and to give a more detailed explanation of the
proofs where I felt that it is useful. (There were many such places.)
In doing so I was influenced by a Russian proverb which says:
`Luchshe vrag khoroshego'. I tried to follow the advice of this
proverb. (I do not know of an English counterpart of it, but it has
a French version: `Le mieux est l'ennemi du bien'.)

I made only one exception. I decided to explain those basic notions
and results in the theory of generalized functions which were applied
in this work in an implicit way. In particular, I tried to explain
how one gets with the help of this theory those results about the
so-called spectral representation of the covariance function of
stationary random fields that I have formulated under the name
{\it Bochner's theorem}\/ and {\it Bochner--Schwartz theorem.} This
extension of the text is contained in the attachments to Sections~1
and~3. In the original version I only referred to a work where
these notions and results can be found. But now I found such an
approach not satisfactory, because these notions and results play
an important role in some arguments of this work. Hence I felt that
to make a self-contained presentation of the subject I have to
explain them in more detail.

\bigskip\noindent
Budapest, 15 August 2011

\medskip
\hskip8truecm P\'eter Major

\vfill\eject

\pageno=1

\beginsection 1. On a limit problem.

We begin with the formulation of a problem which is important both for
probability theory and statistical physics. The multiple Wiener--It\^o
integral proved to be a very useful tool at the investigation of this
problem.

We shall consider a set of random variables $\xi_n$,
$n\in \text{\BBB Z}_\nu$, where $\text{\BBB Z}_\nu$ denotes the
$\nu$-dimensional integer lattice, and we shall study their properties.
Such a set of random variables will be called a discrete
($\nu$-dimensional) random field. We shall be mainly interested in
so-called stationary random fields. Let us recall their definition.

\medskip\noindent
{\bf Definition of discrete (strictly) stationary random fields.} {\it
A set of random variables $\xi_n$, $n\in\text{\BBB Z}_\nu$, is
called a (strictly) stationary discrete random field if
$(\xi_{n_1},\dots,\xi_{n_k})\overset\Delta\to=
(\xi_{n_1+m},\dots,\xi_{n_k+m})$
for all $k=1,2,\dots$ and \ $n_1,\dots,n_k,\;m\in\text{\BBB Z}_\nu$,
where $\overset\Delta\to=$ denotes equality in distribution.}

\medskip
Let us also recall that a discrete random field $\xi_n$,
$n\in\text{\BBB Z}_\nu$, is called Gaussian if for every finite
subset $\{n_1,\dots,n_k\}\subset\text{\BBB Z}_\nu$ the random
vector $(\xi_{n_1},\dots,\xi_{n_k})$ is normally distributed.

Given a discrete random field $\xi_n$, $n\in\text{\BBB Z}_\nu$, we
define for all $N=1,2,\dots$ the new random fields
$$
Z_n^N=A_N^{-1}\sum_{j\in B_n^N}\xi_j, \qquad N=1,2,\dots, \quad
n\in\text{\BBB Z}_\nu,
\tag1.1
$$
where
$$
B_n^N=\{j\colon\; j\in \text{\BBB Z}_\nu,\quad n^{(i)}N\le
j^{(i)}<(n^{(i)}+1)N,\;i=1,2,\dots, \nu\},
$$
and $A_N$, $A_N>0$, is an appropriate norming constant. The superscript
$i$ denotes the $i$-th coordinate of a vector in this formula. We are
interested in the question when the finite dimensional distribution of
the random fields $Z_n^N$ defined in~(1.1) have a limit as $N\to\infty$.
In particular, we would like to describe those random fields $Z_n^*$,
$n\in\text{\BBB Z}_\nu$, which appear as the limit of such random
fields~$Z_n^N$. This problem led to the introduction of the following
notion.

\medskip\noindent
{\bf Definition of self-similar (discrete) random fields.} {\it A
(discrete) random field $\xi_n$, $n\in\text{\BBB Z}_\nu$, is called
self-similar with self-similarity parameter~$\alpha$ if the random
fields $Z^N_n$ defined in~(1.1) with their help and the choice
$A_N=N^\alpha$ satisfy the relation
$$
(\xi_{n_1},\dots,\xi_{n_k})\overset\Delta\to=(Z^N_{n_1},\dots,Z_{n_k}^N)
\tag1.2
$$
for all $N=1,2,\dots$ and $n_1,\dots,n_k\in\text{\BBB Z}_\nu$.}

\medskip
We are interested in the choice $A_N=N^\alpha$ with some $\alpha>0$
in the definition of the random variables~$Z^N_n$ in~(1.2), because
under slight restrictions, relation~(1.2) can be satisfied only with
such norming constants $A_N$. A central problem both in statistical
physics and in probability theory is the description of self-similar
fields. We are interested  in self-similar fields whose random
variables have a finite second moment. This excludes the fields
consisting of i.i.d. random variables with a non--Gaussian stable law.

The Gaussian self-similar fields and their Gaussian range of attraction
are fairly well known. Much less is known about the non-Gaussian case.
The problem is hard, because the transformations of measures over
$R^{\text{\BBB Z}_\nu}$ induced by formula~(1.1) have a very
complicated structure. We shall define the so-called subordinated
fields below. (More precisely the fields subordinated to a stationary
Gaussian field.) In case of subordinated fields the Wiener--It\^o
integral is a very useful tool for investigating the transformation
defined in~(1.1). In particular, it enables us to construct
non--Gaussian self-similar fields and to prove non-trivial limit
theorems. All known results are closely related to this technique.

Let $X_n$, $n\in\text{\BBB Z}_\nu$, be a stationary Gaussian field. We
define the shift transformations $T_m$, $m\in\text{\BBB Z}_\nu$, over
this field by the formula $T_mX_n=X_{n+m}$ for all
$n,\,m\in\text{\BBB Z}_\nu$. Let $\Cal H$ denote the {\it real} Hilbert
space consisting of the square integrable random variables measurable
with respect to the $\sigma$-algebra
$\Cal B=\Cal B(X_n,\;n\in\text{\BBB Z}_\nu)$. The scalar product in
$\Cal H$ is defined as $(\xi,\eta)=E\xi\eta$, $\xi,\,\eta\in\Cal H$.
The shift transformations $T_m$, $m\in\text{\BBB Z}_\nu$, can be
extended to a group of unitary shift transformations over $\Cal H$ in
a natural way. Namely, if $\xi=f(X_{n_1},\dots,X_{n_k})$ then we
define $T_m\xi=f(X_{n_1+m},\dots,X_{n_k+m})$. It can be seen that
$\|\xi\|=\|T_m\xi\|$, and the above considered random variables $\xi$
are dense in $\Cal H$. (A more detailed discussion about the definition
of shift operators and their properties will be given in Section~2 in
a {\it Remark}\/ after the formulation of Theorem~2C. Here we shall
define the shift $T_m\xi$, $m\in\text{\BBB Z}_\nu$, of all random
variables $\xi$ which are measurable with respect to the
$\sigma$-algebra $\Cal B(X_n,\,n\in{\BBB Z}_\nu)$, i.e. $\xi$ does
not have to be square integrable.) Hence $\|T_m\|$ can be extended
to the whole space $\Cal H$ by $L_2$ continuity. It can be proved
that the norm preserving transformations $T_m$, $m\in\text{\BBB Z}_\nu$,
constitute a unitary group in $\Cal H$, i.e. $T_{n+m}=T_nT_m$ for
all $n,\,m\in\text{\BBB Z}_\nu$, and $T_0=\text{Id}$. Now we
introduce the following

\medskip\noindent
{\bf Definition of subordinate fields.} {\it Given a stationary Gaussian
field $X_n$, $n\in\text{\BBB Z}_\nu$, we define the Hilbert spaces
$\Cal H$ and the shift transformations $T_m$, $m\in\text{\BBB Z}_\nu$,
over $\Cal H$ as before. A discrete stationary field $\xi_n$ is called
a random field subordinated to $X_n$ if $\xi_n\in\Cal H$, and
$T_n\xi_m=\xi_{n+m}$ for all $n,\,m\in\text{\BBB Z}_\nu$.}

\medskip
We remark that $\xi_0$ determines the subordinated fields $\xi_n$
completely, since $\xi_n=T_n\xi_0$. Later we give a more adequate
description of subordinates fields by means of Wiener--It\^o integrals.
Before working out the details we formulate the continuous time
version of the above notions and problems. In the continuous time
case it is more natural to consider generalized random fields. To
explain the idea behind such an approach we shortly explain a
different but equivalent description of discrete random fields.
We present them as an appropriate set of random variables indexed
by the elements of a linear space. This shows some similarity with
generalized random fields.

Let $\varphi_n(x)$, $n\in\text{\BBB Z}_\nu$, $n=(n_1,\dots,n_\nu)$,
denote the indicator function of the cube
$[n_1-\frac12,n_1+\frac12)\times\cdots\times[n_\nu-\frac12,n_\nu+\frac12)$,
with center $n=(n_1,\dots,n_\nu)$ and with edges of length~1,
ie.\ let $\varphi_n(x)=1$, $x=(x_1,\dots,x_\nu)\in R^\nu$, if
$n_j-\frac12\le x_j<n_j+\frac12$ for all $1\le j\le \nu$, and let
$\varphi_n(x)=0$ otherwise. Define the linear space $\Phi$ of
functions on $R^\nu$ consisting of all finite linear combinations of
the form $\sum c_j\varphi_{n_j}(x)$, $n_j\in\text{\BBB Z}_\nu$, with
the above defined functions $\varphi_n(x)$ and real
coefficients~$c_j$. Given a discrete random field $\xi_n$,
$n\in\text{\BBB Z}_\nu$, define the random variables $\xi(\varphi)$
for all $\varphi\in\Phi$ in the following way. Put
$\xi(\varphi)=\sum c_j \xi_{n_j}$ if
$\varphi(x)=\sum c_j\varphi_{n_j}(x)$. In particular,
$\xi(\varphi_n)=\xi_n$ for all~$n\in\text{\BBB Z}_\nu$. The
identity $\xi(c_1\varphi+c_2\psi)=c_1\xi(\varphi)+c_2\xi(\psi)$
also holds for all $\varphi,\psi\in\Phi$ and real numbers $c_1$
and~$c_2$.

Let us also define the function
$\varphi^{(N,A_N)}(x)=\frac1{A(N)}\varphi(\frac xN)$ for all
functions $\varphi\in\Phi$ and positive integers $N=1,2,\dots$,
with some appropriately chosen constants $A_N>0$. Observe that
$\xi(\varphi^{(N,A_N)}_n)=Z^N_n$
with the random variable $Z^N_n$ defined in~(1.1). All previously
introduced notions related to discrete random fields can be
reformulated with the help of the set of random variables
$\xi(\varphi)$, $\varphi\in\Phi$. Thus for instance the random
field $\xi_n$, $n\in\text{\BBB Z}_\nu$ is self-similar with
self-similarity parameter~$\alpha$ if and only if
$\xi(\varphi^{(N,N^\alpha)})\overset\Delta\to=\xi(\varphi)$
for all $\varphi\in\Phi$ and $N=1,2,\dots$. (To see why this
statement holds observe that the distributions of two random
vectors agree if and only if every linear combination of their
coordinates have the same distribution. This follows from the fact
that the characteristic function of a random vector determines its
distribution.)

It will be more useful to define the continuous time version of
discrete random fields as generalized random fields. The generalized
random fields will be defined as a set of random variables indexed
by the elements of a linear space of functions. They show some
similarity to the class of random variables $\xi(\varphi)$,
$\varphi\in\Phi$, defined above. The main difference is that instead
of the space~$\Phi$ a different linear space is chosen for the
parameter set of the random field. We shall choose the so-called
Schwartz space for this role.

Let $\Cal S=\Cal S_\nu$ be the Schwartz space of (real valued) rapidly
decreasing, smooth functions on $R^\nu$. (See e.g.~[15] for the
definition of $\Cal S_\nu$. I shall present a more detailed discussion
about the definition of the space $\Cal S$ in the adjustment to
Section~1.)  Generally one takes the space of complex valued, rapidly
decreasing, smooth functions as the  space~$\Cal S$, but we shall
denote the space of {\it real valued},\/ rapidly decreasing, smooth
functions by~$\Cal S$ if we do not say this otherwise. We shall omit
the subscript $\nu$ if it leads to no ambiguity. Now we introduce the
notion of generalized random fields.

\medskip\noindent
{\bf Definition of generalized random fields.} {\it We say that the set
of random variables $X(\varphi)$,  $\varphi\in\Cal S$, is a generalized
random field over the Schwartz space $\Cal S$ of rapidly decreasing,
smooth functions if:

\medskip
\item{a)} $X(a_1\varphi_1+a_2\varphi_2)=a_1X(\varphi_1)+a_2X(\varphi_2)$
with probability 1 for all real numbers $a_1$ and $a_2$ and
$\varphi_1\in\Cal S$, $\varphi_2\in\Cal S$. (The exceptional set of
probability~0 where this identity does not hold may depend on $a_1$,
$a_2$, $\varphi_1$ and $\varphi_2$.)
\item{b)} $X(\varphi_n)\Rightarrow X(\varphi)$ stochastically if
$\varphi_n\to\varphi$ in the topology of $\Cal S$.}

\medskip
We also introduce the following definitions.

\medskip\noindent
{\bf Definition of stationarity and Gaussian property of a generalized
random field. On the notion of convergence of generalized random fields
in distribution.} {\it The generalized random field
$X=\{X(\varphi),\,\varphi\in \Cal S\}$ is stationary if
$X(\varphi)\overset\Delta\to=X(T_t\varphi)$ for all $\varphi\in\Cal S$
and $t\in R^\nu$, where $T_t\varphi(x)=\varphi(x-t)$. It is Gaussian
if $X(\varphi)$ is a Gaussian random variable for all
$\varphi\in\Cal S$. The relation
$X_n\overset{\Cal D}\to\rightarrow X_0$ as $n\to\infty$ holds for a
sequence of generalized random fields $X_n$, $n=0,1,2,\dots$, if
$X_n(\varphi)\overset{\Cal D}\to\rightarrow X_0(\varphi)$ for all
$\varphi\in\Cal S$, where
$\overset{\Cal D}\to\rightarrow$ denotes convergence in distribution.}

\medskip
Given a stationary generalized  random field $X$ and a function
$A(t)>0$, $t>0$, on the set of positive real numbers we define the
(stationary) random fields $X^A_t$ for all $t>0$ by the formula
$$
X^A_t(\varphi)=X(\varphi_t^A), \quad \varphi\in\Cal S, \qquad
\text{where } \varphi_t^A(x)=A(t)^{-1}\varphi\(\frac xt\). \tag1.3
$$

We are interested in the following
 
\medskip\noindent
{\bf Question.} {\it When does a generalized random field $X^*$ exist such
that $X_t^A\overset{\Cal D}\to\rightarrow X^*$ as $t\to\infty$ (or as
$t\to0$)?}

\medskip
In relation to this question we introduce the following

\medskip\noindent
{\bf Definition of self-similarity.} {\it The stationary generalized
random field $X$ is self-similar with self-similarity parameter $\alpha$
if  $X^A_t(\varphi)\overset\Delta\to= X(\varphi)$ for all
$\varphi\in\Cal S$  and $t>0$ with the function $A(t)=t^\alpha$.}

\medskip
To answer the above question one should first describe the generalized
self-similar random fields.

We try to explain the motivation behind the above definitions. Given
an ordinary random field $X(t)$, $t\in R^\nu$, and a topological
space $\Cal E$ consisting of functions over $R^\nu$ one can define
the random variables $X(\varphi)=\int_{R^\nu} \varphi(t)X(t)\,dt$,
$\varphi\in\Cal E$. Some difficulty may arise when defining this
integral, but it can be overcome in all interesting cases. If the
space $\Cal E$ is rich enough, and this is the case if
$\Cal E=\Cal S$, then the integrals $X(\varphi)$, $\varphi\in\Cal E$,
determine the random process $X(t)$. The set of random variables
$X(\varphi)$, $\varphi\in\Cal S$, is a generalized random field
in all nice cases. On the other hand, there are generalized random
fields which cannot be obtained by integrating ordinary random fields.
In particular, the generalized self-similar random fields we shall
construct later cannot be interpreted through ordinary fields. The
above definitions of various properties of generalized fields are
fairly natural, considering what these definitions mean for
generalized random fields obtained by integrating ordinary fields.

The investigation of generalized random fields is simpler than that
of ordinary discrete random fields, because in the continuous case
more symmetry is available. Moreover, in the study or construction
of discrete random fields generalized random fields may play a useful
role. To understand this let us remark that if we have a generalized
random field $X(\varphi)$, $\varphi\in\Cal S$, and we can extend the
space $\Cal S$ containing the test function $\varphi$ to such a larger
linear space $\Cal T$ for which $\Phi\subset\Cal T$ with the above
introduced linear space~$\Phi$, then we can define the discrete random
field $X(\varphi)$, $\varphi\in\Phi$, by a restriction of the space of
test functions of the generalized random field $X(\varphi)$,
$\varphi\in\Cal T$. This random field can be considered as the
discretization of the  original generalized random field
$X(\varphi)$, $\varphi\in\Cal S$.

We finish this section by defining the generalized subordinated
random fields. Let $X(\varphi)$, $\varphi\in\Cal S$, be a
generalized stationary Gaussian random field. The formula
$T_tX(\varphi))=X(T_t\varphi)$, $T_t\varphi(x)=\varphi(x-t)$,
 defines the shift transformation for all $t\in R^\nu$. Let
$\Cal H$ denote the real Hilbert space consisting of the
$\Cal B=\Cal B(X(\varphi),\;\varphi\in\Cal S)$ measurable random
variables with finite second moment. The shift transformation can
be extended to a group of unitary transformations over $\Cal H$
similarly to the discrete case.

\medskip\noindent
{\bf Definition of generalized random fields subordinated to a
generalized stationary Gaussian random field.} {\it Given a
generalized stationary Gaussian random field $X(\varphi)$,
$\varphi\in\Cal S$, we define the Hilbert space $\Cal H$ and the
shift transformations $T_t$, $t\in R^\nu$, over $\Cal H$ as
above. A generalized stationary random field $\xi(\varphi)$,
$\varphi\in\Cal S$, is subordinated to the field $X(\varphi)$,
$\varphi\in\Cal S$, if $\xi(\varphi)\in\Cal H$ and
$T_t\xi(\varphi)=\xi(T_t\varphi)$ for all $\varphi\in\Cal S$ and
$t\in R^\nu$, and $E[\xi\varphi_n)-\xi(\varphi)]^2\to0$ if
$\varphi_n\to\varphi$ in the topology of $\Cal S$.}

\medskip\noindent
{\bf Attachment to Section 1.} {\it A brief overview about some
results on generalized functions.}

\medskip\noindent
Let us first describe the Schwartz spaces $\Cal S$ and $\Cal S^c$ in
more detail. The space $\Cal S^c=(\Cal S_\nu)^c$ consists of those
complex valued functions of $\nu$ variables which decrease at
infinity, together with their derivatives, faster than any polynomial
degree. More explicitly, $\varphi\in\Cal S^c$ for a complex valued
function $\varphi$ of $\nu$ variables if
$$
\left|x_1^{k_1}\cdots x_\nu^{k_\nu}\frac{\partial^{q_1+\cdots+q_\nu}}
{\partial x_1^{q_1}\dots \partial x_\nu^{q_\nu}}
\varphi(x_1,\dots,x_\nu)\right|
\le C(k_1,\dots,k_\nu,q_1,\dots,q_\nu)
$$
for all point $x=(x_1,\dots,x_\nu)\in R^\nu$ and  vectors
$(k_1,\dots,k_\nu)$, $(q_1,\dots,q_\nu)$ with non-negative
integer coordinates with some constant
$C(k_1,\dots,k_\nu,q_1,\dots,q_\nu)$ which may depend on the
function~$\varphi$. This formula can be written in a more concise
form as
$$
|x^k D^q\varphi(x)|\le C(k,q) \quad \text{with } k=(k_1,\dots,k_\nu)
 \text{ and } q=(q_1,\dots,q_\nu),
$$
where $x=(x_1,\dots,x_\nu)$, $x^k=x_1^{k_1}\cdots x_\nu^{k_\nu}$ and
$D^q=\frac{\partial^{q_1+\cdots+q_\nu}}
{\partial x_1^{q_1}\dots \partial x_\nu^{q_\nu}}$.
The elements of the space $\Cal S$ are defined
similarly, with the only difference that they are real valued functions.

To define the spaces $\Cal S$ and $\Cal S^c$ we still have to define
the convergence in them. We say that a sequence of functions
$\varphi_n\in\Cal S^c$ (or $\varphi_n\in\Cal S$) converges to a function
$\varphi$ if
$$
\lim_{n\to\infty}\sup_{x\in R^\nu}
(1+|x|^2)^k|D^q\varphi_n(x)-D^q\varphi(x)|=0.
$$
for all $k=1,2,\dots$ and $q=(q_1,\dots,q_\nu)$.
It can be seen that the limit function $\varphi$ is also in the
space~$\Cal S^c$ (or in the space $\Cal S$).

A nice topology can be introduced in the space $\Cal S^c$ (or $\Cal S$)
which induces the above convergence. The following topology is an
appropriate choice. Let a basis of neighbourhoods of the origin consist
of the sets
$$
U(k,q,\e)=\left\{\varphi\colon\;\max_x(1+|x|^2)^k |D^q\varphi(x)|<\e\right\}
$$
with $k=0,1,2,\dots$, $q=(q_1,\dots,q_\nu)$ with non-negative integer
coordinates and $\e>0$, where $|x|^2=x_1^2+\cdots+x_\nu^2$. A basis
of neighbourhoods of an arbitrary function $\varphi\in\Cal S^c$ (or
$\varphi\in\Cal S$) consists of sets of the form $\varphi+U(k,q,\e)$,
where the class of sets $U(k,q,\e)$ is a basis of neighbourhood of
the origin. The fact that the convergence in $\Cal S$ has such a
representation, (and a similar result holds in some other spaces
studied in the theory of generalized functions) has a great
importance in the theory of generalized functions. We also have
exploited this fact in Section~6 of this Lecture Note. Topological
spaces with such a topology are called countably normed spaces.

The space of generalized functions $\Cal S'$ consists of the
{\it continuous}\/ linear maps $F\colon\;\Cal S\to C$ or
$F\colon\;\Cal S^c\to C$, where $C$ denotes the linear space of
complex numbers. (In the study of the space $\Cal S'$ we omit
the upper index~$c$, i.e. we do not indicate whether we are working in
real or complex space when this causes no problem.) We shall write the
map $F(\varphi)$, $F\in\Cal S'$ and $\varphi \in\Cal S$ (or
$\varphi\in\Cal S^c$) in the form~$(F,\varphi)$.

We can define generalized functions $F\in\Cal S'$ by the formula
$(F,\varphi)=\int \overline{f(x)}\varphi(x)\,dx$ for all
$\varphi\in\Cal S$ or $\varphi\in\Cal S^c$ with a function $f$ such
that $\int(1+|x|^2)^{-p}|f(x)|\,dx<\infty$ with some $p\ge0$.
(The upper script~$\bar{}$ denotes complex conjugate in the sequel.)
Such functionals are called regular. There are also non-regular
functionals in the space~$\Cal S'$. An example for them is the
$\delta$-function defined by the formula
$(\delta ,\varphi)=\varphi(0)$. There is a rather good description of
the generalized functions $F\in\Cal S'$, (see the book I.~M.~Gelfand
and G. E.~Shilov: Generalized functions, Volume~2, Chapter~2,
Section~4), but we do not need this result, hence we do not discuss
it here. Another important question in this field that we omit is
about the interpretation of a usual function as  a generalized
function in the case when it does not define a regular functional
because of its strong singularity in some points. In such cases some
regularization can be applied. It is an important problem to find the
appropriate generalized functions in such cases, but it does not
appear in the study of the problems of this work.

The derivative and the Fourier transform of generalized functions are
also defined, and they play an important role in some investigations.
In the definition of these notions for generalized functions we want
to preserve the old definition if nice regular functionals are
considered for which these notions were already defined in classical
analysis. Such considerations lead to the definition
$(\frac{\partial_j}{\partial x_j}F,\varphi)
=-(F,\frac{\partial\varphi}{\partial x_j})$  of the derivative of
generalized functions. We do not discuss this definition in more
detail, because here we do not work with the derivatives of
generalized functions.

The Fourier transform of generalized functions in~$S'$ appears in
our discussion, although only in an implicit form. The
Bochner-Schwartz theorem discussed in Section~3 actually deals with
the Fourier transform of generalized functions. Hence the definition
of Fourier transform will be given in more detail.

We shall define the Fourier transform of a generalized function by
means of a natural extension of the Parseval formula, more explicitly
of a simplified version of it, where the same identity
$$
\int_{R^\nu} \overline{f(x)}g(x)\,dx
=\frac1{(2\pi)^\nu}\int_{R^\nu} \overline{\tilde f(u)}\tilde g(u)\,du
$$
is formulated with $\tilde f(u)=\int_{R^\nu}e^{i(u,x)}f(x)\,dx$
and $\tilde g(u)=\int_{R^\nu}e^{i(u,x)}g(x)\,dx$. But now we
consider a pair of functions $(f,g)$ with different properties.
We demand that $f$ should be an integrable function, and
$g\in\Cal S^c$. (In the original version of the Parseval formula
both~$f$ and~$g$ are $L_2$ functions.)

The proof of this identity is simple. Indeed, since the function
$g\in\Cal S^c$ can be calculated as the inverse Fourier transform
of its Fourier transform~$\tilde g\in\Cal S^c$, i.e.\
$g(x)=\frac1{(2\pi)^\nu}\int e^{-i(u,x)}\tilde g(u)\,du$, we can
write
$$
\align
\int \overline{f(x)}g(x)\,dx
&=\int \overline{f(x)}\[\frac1{(2\pi)^\nu}\int e^{-i(u,x)}
\tilde g(u)\,du\]\,dx\\
&=\int\tilde g(u)\[\frac1{(2\pi)^\nu}\int\overline{ e^{i(u,x)}f(x)}\,dx\]\,du
=\frac1{(2\pi)^\nu}\int \overline{\tilde f(u)}\tilde g(u)\,du.
\endalign
$$
Let us also remark that the Fourier transform $f\to\tilde f$ is a
bicontinuous map from $\Cal S^c$ to~$\Cal S^c$. (This means that this
transformation is invertible, and both the Fourier transform and
its inverse are continuous maps from $\Cal S^c$ to $\Cal S^c$.) (The
restriction of the Fourier transform to the space $\Cal S$ of real
valued functions is a bicontinuous map from $\Cal S$ to the subspace
of $\Cal S^c$ consisting of those functions $f\in\Cal S^c$ for which
$f(-x)=\overline{f(x)}$ for all $x\in R^\nu$.)

The above results make natural the following definition of the Fourier
transform~$\tilde F$ of a generalized function $F\in\Cal S'$.
$$
(\tilde F, \tilde\varphi)=(2\pi)^\nu(F,\varphi)
\quad\text{for all } \varphi\in\Cal S^c.
$$
Indeed, if $F\in\Cal S'$ then $\tilde F$ is also a continuous linear map
on $\Cal S^c$, i.e. it is also an element of~$\Cal S'$. Beside this,
the above proved version of the Parseval formula implies that
if we consider an integrable function~$f$ on~$R^\nu$ both as a usual
function and as a (regular) generalized function, its Fourier
transform agrees in the two cases.

\medskip
There are other classes of test functions and spaces of generalized
functions studied in the literature. The most popular among them is
the space~$\Cal D$ of infinitely many differentiable functions with
compact support and its dual space~$\Cal D'$, the space of continuous
linear transformations on the space $\Cal D$. (These spaces are
generally denoted  by~$\Cal D$ and~$\Cal D'$ in the literature,
although just the book~[15] that we use as our main reference
in this subject applies the notation $\Cal K$ and~$\Cal K'$ for
them.) We shall discuss this space only very briefly.

The space $\Cal D$ consists of the infinitely many times
differentiable functions with compact support. Thus it is a
subspace of~$\Cal S$. A sequence $\varphi_n\in\Cal D$, $n=1,2,\dots$,
converges to a function~$\varphi$, if there is a compact set
$A\subset R^\nu$ which is the support of all these
functions~$\varphi_n$, and the functions $\varphi_n$ together with
all their derivatives converge uniformly to the
function~$\varphi$ and to its corresponding derivatives. It is not
difficult to see that also $\varphi\in\Cal D$, and if the functions
$\varphi_n$ converge to $\varphi$ in the space~$\Cal D$, then they
also converge to~$\varphi$ in the space~$\Cal S$. Moreover, $\Cal D$
is an everywhere dense subspace of~$\Cal S$. The space
$\Cal D'$ consists of the continuous linear functionals in~$\Cal D$.

The results describing the behaviour of $\Cal D$ and $\Cal D'$ are
very similar to those describing the behaviour of $\Cal S$
and~$\Cal S'$. There is one difference that deserves some attention.
The Fourier transforms of the functions in $\Cal D$ may not belong
to~$\Cal D$. The class of these Fourier transforms can be described
by means of some results in complex analysis. A topological
space~$\Cal Z$ can be defined on the set of Fourier transforms of
the functions from the space~$\Cal D$. If we want to apply Fourier
analysis in the space~$\Cal D$, then we also have to study this
space~$\Cal Z$ and its dual space~$\Cal Z'$. I omit the details.

\beginsection 2. Wick polynomials.

In this section we consider the so-called Wick polynomials, a
multi-dimensional generalization of Hermite polynomials. They are
closely related to multiple Wiener--It\^o integrals.

Let $X_t$, $t\in T$, be a set of jointly Gaussian random variables
indexed by a parameter set $T$. Let $EX_t=0$ for all $t\in T$. We
define the real Hilbert space $\Cal H_1$ and $\Cal H$ in the following
way: A square integrable random variable is in $\Cal H$ if and only if
it is measurable with respect to the $\sigma$-algebra
$\Cal B=\Cal B(X_t,\;t\in T)$, and the scalar product in $\Cal H$ is
defined as $(\xi,\eta)=E\xi\eta$, $\xi,\,\eta\in\Cal H$. The Hilbert
space $\Cal H_1\subset\Cal H$ is the subspace of $\Cal H$ generated
by the finite linear combinations $\sum c_jX_{t_j}$, $t_j\in T$. We
consider only such sets of Gaussian random variables $X_t$ for which
$\Cal H_1$ is separable. Otherwise $X_t$, $t\in T$, can be arbitrary,
but the most interesting case for us is when $T=\Cal S_\nu$ or
$\text{\BBB Z}_\nu$, and $X_t$, $t\in T$, is a stationary Gaussian
field.

Let $Y_1,Y_2,\dots$ be an orthonormal basis in $\Cal H_1$. The
uncorrelated random variables $Y_1,Y_2,\dots$ are independent, since
they are (jointly) Gaussian. Moreover,
$\Cal B(Y_1,Y_2,\dots)=\Cal B(X_t,\;t\in T)$.  Let $H_n(x)$ denote the
$n$-th Hermite polynomial with leading coefficient~1, i.e. let
$H_n(x)=(-1)^ne^{x^2/2}\frac{d^n}{dx^n}(e^{-x^2/2})$. We recall the
following results from analysis and measure theory.

\medskip\noindent
{\bf Theorem 2A.} {\it The Hermite polynomials $H_n(x)$, $n=0,1,2,\dots$,
form a complete orthogonal system in
$L_2\(R,\Cal B,\frac1{\sqrt{2\pi}}e^{-x^2/2}\,dx\)$. (Here $\Cal B$
denotes the Borel $\sigma$-algebra on the real line.)}

\medskip
Let  $(X_j,\Cal X_j,\mu_j)$, $j=1,2,\dots$, be countably many independent
copies of a probability space $(X,\Cal X,\mu)$. (We denote the points
of $X_j$ by $x_j$.) Let $(X^\infty,\Cal X^\infty,\mu^\infty)
=\prodd_{j=1}^\infty(X_j,\Cal X_j,\mu_j)$. With such a notation the
following result holds.

\medskip\noindent
{\bf Theorem 2B.} {\it Let $\varphi_0,\varphi_1,\dots$,
$\varphi_0(x)\equiv1$, be a complete orthonormal system in the
Hilbert space $L_2(X,\Cal X,\mu)$. Then the functions
$\prodd_{j=1}^\infty\varphi_{k_j}(x_j)$, where only finitely many
indices $k_j$ differ from~0, form a complete orthonormal basis in
$L_2(X^\infty,\Cal X^\infty,\mu^\infty)$.}

\medskip\noindent
{\bf Theorem 2C.} {\it Let $Y_1,Y_2,\dots$ be random variables on
a probability space $(\Omega,\Cal A,P)$ taking values in a measurable
space  $(X,\Cal X)$. Let $\xi$ be a real valued random variable
measurable with respect to the $\sigma$-algebra $\Cal B(Y_1,Y_2,\dots)$,
and let $(X^\infty,\Cal X^\infty)$ denote the infinite product
$(X\times X\times\cdots,\Cal X\times \Cal X\times\cdots)$ of the space
$(X,\Cal X)$ with itself. Then there exists a real valued, measurable
function~$f$ on the space $(X^\infty,\Cal X^\infty)$ such that
$\xi=f(Y_1,Y_2,\dots)$.}

\medskip\noindent
{\it Remark.}\/ Let us have a stationary random field $X_n(\oo)$,
$n\in\text{\BBB Z}_\nu$. Theorem~2C enables us to extend the shift
transformation $T_m$, defined as $T_mX_n(\oo)=X_{n+m}(\oo)$,
$n,\,m\in\text{\BBB Z}_\nu$, for all random variables $\xi(\oo)$,
measurable with respect to the $\sigma$-algebra
$\Cal B(X_n(\oo),\,n\in\text{\BBB Z}_\nu)$. Indeed, by
Theorem~2C we can write $\xi(\oo)=f(X_n(\oo),\,n\in\text{\BBB Z}_\nu)$,
and define $T_m\xi(\oo)=f(X_{n+m}(\oo),\,n\in\text{\BBB Z}_\nu)$. We
still have to understand, that although the function~$f$ is not unique
in the representation of the random variable~$\xi(\oo)$, the above
definition of $T_m\xi(\oo)$ is meaningful. To see this we have to
observe that if
$f_1(X_n(\oo),\,n\in\text{\BBB Z}_\nu)=f_2(X_n(\oo),\,n\in\text{\BBB Z}_\nu)$
for two functions $f_1$ and $f_2$ with probability~1, then also
$f_1(X_{n+m}(\oo),\,n\in\text{\BBB Z}_\nu)
=f_2(X_{n+m}(\oo),\,n\in\text{\BBB Z}_\nu)$
with probability~1 because of the stationarity of the random
field~$X_n(\oo)$, $n\in\text{\BBB Z}_\nu$. Let us also observe that
$\xi(\oo)\overset\Delta\to=T_m\xi(\oo)$ for all $m\in\text{\BBB Z}_\nu$.
Beside this, $T_m$ is a linear operator on the linear space of random
variables, measurable with respect to the $\sigma$-algebra
$\Cal B(X_n,\,n\in\text{\BBB Z}_\nu)$. If we restrict it to the space
of square integrable random variables, then $T_m$ is a unitary operator,
and the operators $T_m$, $m\in\text{\BBB Z}_\nu$, constitute a unitary
group.

Let a stationary generalized field $X=\{X(\varphi),\,\varphi\in\Cal S\}$
be given. The shift $T_t\xi$ of a random variable $\xi$, measurable with
respect to the $\sigma$-algebra $\Cal B(X(\varphi),\,\varphi\in\Cal S)$
can be defined for all $t\in R^\nu$ similarly to the discrete case with
the help of Theorem~2C and the following observation: If
$\xi\in\Cal B(X(\varphi),\,\varphi\in\Cal S)$ for a random
variable~$\xi$, then there exists such a countable subset
$\{\varphi_1,\varphi_2,\dots\}\subset\Cal S$ (depending on the random
variable~$\xi$) for which $\xi$ is
$\Cal B(X(\varphi_1),X(\varphi_2),\dots)$ measurable. (We write
$\xi(\oo)=f(X(\varphi_1)(\oo),X(\varphi_2)(\oo),\dots)$ with
appropriate functions $f$, and $\varphi_1\in\Cal S$,
$\varphi_2\in\Cal S$,\dots, and define the shift $T_t\xi$ as
$T_t\xi(\oo)=f(X(T_t\varphi_1)(\oo),X(T_t\varphi_2)(\oo),\dots)$,
where $T_t\varphi(x)=\varphi(x-t)$ for $\varphi\in\Cal S$.) The
transformations $T_t$, $t\in R^{\nu}$, are linear operators over the
space of random variables measurable with respect to the
$\sigma$-algebra $\Cal B(X(\varphi),\,\varphi\in\Cal S)$ with similar
properties as their discrete counterpart.

\medskip
Theorems 2A, 2B and 2C have the following important consequence.

\medskip\noindent
{\bf Theorem 2.1.} {\it Let $Y_1,Y_2,\dots$ be an orthonormal basis
in the Hilbert space $\Cal H_1$ defined above with the help of a set
of Gaussian random variables $X_t$, $t\in T$. Then the set of all
possible finite products $H_{j_1}(Y_{l_1})\cdots H_{j_k}(Y_{l_k})$
is a complete orthogonal system in the Hilbert space $\Cal H$
defined above. (Here $H_j(\cdot)$ denotes the $j$-th Hermite
polynomial.)}

\medskip\noindent
{\it The proof of Theorem 2.1.}\/ By Theorems~2A and~2B the set of
all possible products $\prodd_{j=1}^\infty H_{k_j}(x_j)$, where only
finitely many indices $k_j$ differ from 0, is a complete orthonormal
system in $L_2\(R^\infty,\Cal B^\infty,\prodd_{j=1}^\infty
\frac{e^{-x_j^2/2}}{\sqrt{2\pi}}\,dx_j\)$. Since
$\Cal B(X_t,\;t\in T)=\Cal B(Y_1,Y_2,\dots)$, Theorem~2C implies
that the mapping $f(x_1,x_2,\dots,)\to f(Y_1,Y_2,\dots)$ is a unitary
transformation from $L_2\(R^\infty,\Cal B^\infty,\prodd_{j=1}^\infty
\frac{e^{-x_j^2/2}}{\sqrt{2\pi}}\,dx_j\)$ to $\Cal H$. (We call a
transformation from a Hilbert space to another Hilbert space unitary
if it is norm preserving and invertible.) Since the image of a
complete orthogonal system under a unitary transformation is again a
complete orthogonal system, Theorem~2.1 is proved.

\medskip
Let $\Cal H_{\le n}\subset\Cal H$, $n=1,2,\dots$, (with the previously
introduced Hilbert space $\Cal H$) denote the Hilbert space which is the
closure of the linear space consisting of the elements
$P_n(X_{t_1},\dots,X_{t_m})$, where $P_n$ runs through all polynomials
of degree less than or equal to~$n$, and the integer~$m$ and indices
$t_1,\dots,t_m\in T$ are arbitrary. Let $\Cal H_0=\Cal H_{\le 0}$ consist
of the constant functions, and let
$\Cal H_n=\Cal H_{\le n}\ominus\Cal H_{\le n-1}$, $n=1,2,\dots$, where
$\ominus$ denotes orthogonal completion. It is clear that the Hilbert
space $\Cal H_1$ given in this definition agrees with the previously
defined Hilbert space $\Cal H_1$. If $\xi_1,\dots,\xi_m\in\Cal H_1$,
and $P_n(x_1,\dots,x_m)$ is a polynomial of degree $n$, then
$P_n(\xi_1,\dots,\xi_m)\in\Cal H_{\le n}$. Hence Theorem~2.1 implies that
$$
\Cal H=\Cal H_0+\Cal H_1+\Cal H_2+\cdots,  \tag2.1
$$
where $+$ denotes direct sum. Now we introduce the following

\medskip\noindent
{\bf Definition of Wick polynomials.} {\it Given a polynomial
$P(x_1,\dots,x_m)$ of degree~$n$ and a set of (jointly Gaussian)
random variables $\xi_1,\dots,\xi_m\in\Cal H_1$, the Wick polynomial
\hbox{$\:\!P(\xi_1,\dots,\xi_m)\!\:$} is the orthogonal projection
of the random variable $P(\xi_1,\dots,\xi_m)$ to the above defined
subspace $\Cal H_n$ of the Hilbert space $\Cal H$.}

\medskip
It is clear that Wick polynomials of different degree are orthogonal.
Given some $\xi_1,\dots,\xi_m\in\Cal H_1$ define the subspaces
$\Cal H_{\le n}(\xi_1,\dots,\xi_m)\subset\Cal H_{\le n}$, $n=1,2,\dots$,
as the set of all polynomials of the random variables $\xi_1,\dots,\xi_m$
with degree less than or equal to~$n$. Let
$\Cal H_{\le0}(\xi_1,\dots,\xi_m)=\Cal H_{0}(\xi_1,\dots,\xi_m)=\Cal H_0$,
and $\Cal H_n(\xi_1,\dots,\xi_m)=\Cal H_{\le n}(\xi_1,\dots,\xi_m)\ominus
\Cal H_{\le n-1}(\xi_1,\dots,\xi_m)$. With the help of this notation we
formulate the following

\medskip\noindent
{\bf Proposition 2.2.} {\it Let $P(x_1,\dots,x_m)$ be a polynomial of
degree~$n$. Then $\:\!P(\xi_1,\dots,\xi_m)\!\:$ equals the orthogonal
projection of $P(\xi_1,\dots,\xi_m)$ to $\Cal H_n(\xi_1,\dots,\xi_m)$.}

\medskip\noindent
{\it The proof of Proposition 2.2.}\/ Let
$\:\!\bar P(\xi_1,\dots,\xi_m)\!\:$
denote the projection of $P(\xi_1,\dots,\xi_m)$ to
$\Cal H_n(\xi_1,\dots,\xi_m)$. Obviously
$$
P(\xi_1,\dots,\xi_m)-\:\!\bar P(\xi_1,\dots,\xi_m)\!\:\in
\Cal H_{\le n-1}(\xi_1,\dots,\xi_m)\subseteq \Cal H_{\le n-1}.
$$
Hence in order to prove Proposition~2.2 it is enough to show that for all
$\eta\in\Cal H_{\le n-1}$
$$
E\:\!\bar P(\xi_1,\dots,\xi_m)\!\:\eta=0, \tag2.2
$$
since this means that $\:\!\bar P(\xi_1,\dots,\xi_m)\!\:$ is the
orthogonal projection of $P(\xi_1,\dots,\xi_m)\in\Cal H_{\le n}$
to~$\Cal H_{\le n-1}$.

Let $\e_1,\e_2,\dots$ be an orthonormal system in $\Cal H_1$, also
orthonormal to $\xi_1,\dots,\xi_m$, and such that
$\xi_1,\dots,\xi_m,\e_1,\e_2,\dots$ form a basis in $\Cal H_1$.
If $\eta=\prodd_{i=1}^m\xi_i^{l_i}\prodd_{j=1}^\infty\e_j^{k_j}$ with
such exponents $l_i$ and $k_j$ that $\sum l_i+\sum k_j\le n-1$, then
(2.2) holds for this random variable $\eta$ because of the independence
of the random variables $\xi_i$ and $\e_j$. Since the linear
combinations of such $\eta$ are dense in $\Cal H_{\le n-1}$,
formula~(2.2) and Proposition~(2.2) are proved.

\medskip\noindent
{\bf Corollary 2.3.} {\it Let $\xi_1,\dots,\xi_m$ be an orthonormal
system in $\Cal H_1$, and let
$P(x_1,\dots,x_m)=\sum c_{j_1,\dots,j_m}x^{j_1}\cdots x_m^{j_m}$
be a homogeneous polynomial, i.e. let $j_1+\cdots j_m=n$ with some
fixed number~$n$ for all sets $(j_1,\dots,j_m)$ appearing in this
summation. Then
$$
\:\!P(\xi_1,\dots,\xi_m)\!\:=\sum
c_{j_1,\dots,j_m}H_{j_1}(\xi_1)\cdots H_{j_m}(\xi_m).
$$
In particular,
$$
\:\!\xi^n\!\:=H_n(\xi) \quad \text{if } \xi\in\Cal H_1, \text{ and } E\xi^2=1.
$$
}

\medskip\noindent
{\it Remark.} Although we have defined the Wick polynomial (of degree~$n$)
for all polynomials $P(\xi_1,\dots,\xi_m)$ of degree~$n$, we could
have restricted our attention only to homogeneous polynomials of degree
$n$, since the contribution of each terms
$c(j_1,\dots j_m)\xi^{l_1}_1\cdots \xi_m^{l_m}$ of the polynomial
$P(\xi_1,\dots,\xi_m)$ such that $l_1+\cdots+l_m<n$ has a zero
contribution in the definition of the Wick polynomial
$\:\!P(\xi_1,\dots,\xi_m)\!\:$.

\medskip\noindent
{\it Proof of Corollary 2.3.}\/ Let the degree of the polynomial $P$
be~$n$. Then
$$
P(\xi_1,\dots,\xi_m)
-\sum c_{j_1,\dots,j_m}H_{j_1}(\xi_1)\cdots H_{j_m}(\xi_m)
\in \Cal H_{\le n-1}(\xi_1,\dots,\xi_m), \tag2.3
$$
since $P(\xi_1,\dots,x_m)
-\sum c_{j_1,\dots,j_m}H_{j_1}(\xi_1)\cdots H_{j_m}(\xi_m)$
is a polynomial whose degree is less than~$n$. Let
$\eta=\xi_1^{l_1}\cdots\xi_m^{l_m}$, $\summ_{i=1}^ml_i\le n-1$. Then
$$
E\eta H_{j_1}(\xi_1)\cdots H_{j_m}(\xi_m)
=\prod_{i=1}^m E\xi_i^{l_i}H_{j_i}(\xi_i)=0,
$$
since $l_i<j_i$ for at least one index~$i$. Therefore
$$
E\eta\sum c_{j_1,\dots,j_m}H_{j_1}(\xi_1)\cdots H_{j_m}(\xi_m)=0. \tag2.4
$$
Since every element of $\Cal H_{\le n-1}(\xi_1,\dots,\xi_m)$ can be
written as the sum of such elements $\eta$, relation~(2.4) holds for
all $\eta\in\Cal H_{\le n-1}(\xi_1,\dots,\xi_m)$. Relations~(2.3)
and~(2.4) imply Corollary~2.3.

\medskip
The following statement is a simple consequence of the previous results.

\medskip\noindent
{\bf Corollary 2.4.} {\it Let $\xi_1,\xi_2,\dots$ be an orthonormal
basis in $\Cal H_1$. Then the random variables
$H_{j_1}(\xi_1)\cdots H_{j_k}(\xi_k)$, $k=1,2,\dots$,
$j_1+\cdots+j_k=n$, form a complete orthogonal basis in $\Cal H_n$.}

\medskip\noindent
{\it Proof of Corollary 2.4.} It follows from Corollary~2.3 that
$$
H_{j_1}(\xi_1)\cdots H_{j_k}(\xi_k)=\:\!\xi_1^{j_1}\cdots\xi_k^{j_k}\!\:\in
\Cal H_n\quad \text{for all } k=1,2,\dots
$$
if $j_1+\cdots+j_k=n$.
These random variables are orthogonal, and all Wick polynomials
$\:\!P(\xi_1,\dots,\xi_m)\!\:$ of degree~$n$ of the random variables
$\xi_1,\xi_2,\dots$ can be represented as the linear combination of
such terms. Since these Wick polynomials are dense in $\Cal H_n$,
this implies Corollary~2.4.

\medskip
The arguments of this section exploited heavily some properties of
Gaussian random variables. Namely, that the linear combination of Gaussian
random variables is again Gaussian, and in Gaussian case orthogonality
implies independence. This means, in particular, that the rotation of
a standard normal vector leaves its distribution invariant. We finish
this section with an observation based on these facts. This may
illuminate the content of formula~(2.1) from another point of view.
We shall not use the results of the subsequent considerations in the
rest of this work.

Let $U$ be a unitary transformation over $\Cal H_1$. It can be extended
to a unitary transformation $\Cal U$ over $\Cal H$ in a natural way.
Fix an orthonormal basis $\xi_1,\xi_2,\dots$ in $\Cal H_1$, and define
$\Cal U1=1$,
$\Cal U\xi_{j_1}^{l_1}\cdots\xi_{j_k}^{l_k}=(U\xi_{j_1})^{l_1}\cdots
(U\xi_{j_k})^{l_k}$. This transformation can be extended to a linear
transformation $\Cal U$ over $\Cal H$ in a unique way. The transformation
$\Cal U$ is norm preserving, since the joint distributions of
$(\xi_{j_1},\xi_{j_2},\dots)$ and $(U\xi_{j_1},U\xi_{j_2},\dots)$
coincide. Moreover, it is unitary, since $U\xi_1,U\xi_2,\dots$ is
an orthonormal basis in $\Cal H_1$. It is not difficult to see that if
$P(x_1,\dots,x_m)$ is an arbitrary polynomial, and
$\eta_1,\eta_2\dots,\eta_m\in\Cal H_1$, then
$\Cal UP(\eta_1,\dots,\eta_m)=P(U\eta_1,\dots,U\eta_m)$. This relation
means in particular that the transformation $\Cal U$ does not depend on
the choice of the basis in $\Cal H_1$. If the transformations $\Cal U_1$
and $\Cal U_2$ correspond to two unitary transformations $U_1$ and $U_2$
on $\Cal H_1$, then the transformation
$\Cal U_1\Cal U_2$ corresponds to $U_1U_2$. The subspaces $\Cal H_{\le n}$
and therefore the subspaces $\Cal H_n$ remain invariant under the
transformations $\Cal U$.

The shift transformations of a stationary Gaussian field, and their
extensions to $\Cal H$ are the most interesting examples for such
unitary transformations $U$ and $\Cal U$. In the terminology of group
representations the above facts can be formulated in the following
way: The mapping $U\to\Cal U$ is a group representation  of $U(\Cal H_1)$
over $\Cal H$, where $U(\Cal H_1)$ denotes the group of unitary
transformations over $\Cal H_1$. Formula~(2.1) gives a decomposition
of $\Cal H$ into orthogonal invariant subspaces of this representation.

\beginsection 3. Random spectral measures.

Some standard theorems of probability theory state that the correlation
function of a stationary random field can be expressed as the Fourier
transform of a so-called spectral measure. In this section we construct
a random measure with the help of these results, and express the
random field itself as the Fourier transform of this random measure in
some sense. We restrict ourselves to the Gaussian case, although most
of the results in this section are valid for arbitrary stationary
random field with finite second moment if independence is replaced by
orthogonality. In the next section we define the multiple Wiener--It\^o
integrals with respect to this random measure. In the definition of
multiple stochastic integrals the Gaussian property will be heavily
exploited. First we recall two results about the spectral representation
of the covariance function.

Given a stationary Gaussian field $X_n$, $n\in\text{\BBB Z}_\nu$, or
$X(\varphi)$, $\varphi\in\Cal S$, we shall assume throughout the
paper that $EX_n=0$, $EX_n^2=1$ in the discrete and $EX(\varphi)=0$
in the generalized field case.

\medskip\noindent
{\bf Theorem 3A. (Bochner)} {\it Let $X_n$, $n\in\text{\BBB Z}_\nu$, be
a discrete (Gaussian) stationary field. There exists a unique
probability  measure $G$ on $[-\pi,\pi)^\nu$ such that the correlation
function $r(n)=EX_0X_n=EX_kX_{k+n}$, $n\in\text{\BBB Z}_\nu$,
$k\in \text{\BBB Z}_\nu$, can be written in the form
$$
r(n)=\int e^{i(n,x)}G(\,dx), \tag3.1
$$
where $(\cdot,\cdot)$ denotes scalar product. Further $G(A)=G(-A)$
for all $A\in [-\pi,\pi)^\nu$.}

\medskip
We can identify $[-\pi,\pi)^\nu$ with the torus
$R^\nu/2\pi\text{\BBB Z}_\nu$. Thus
e.g. $-(-\pi,\dots,-\pi)=(-\pi,\dots,-\pi)$.

\medskip\noindent
{\bf Theorem 3B. (Bochner--Schwartz)} {\it Let $X(\varphi)$,
$\varphi\in\Cal S$, be a generalized (Gaussian) stationary random
field over $\Cal S=\Cal S_\nu$, which satisfies the condition
$E(X(\varphi_n)-X(\varphi))^2\to0$ if $\varphi_n\to\varphi$ in
the topology of the Schwartz space $\Cal S$. There exists a unique
$\sigma$-finite measure $G$ on $R^\nu$ such that
$$
EX(\varphi)X(\psi)=\int\tilde\varphi(x)\,\bar{\!\tilde\psi}(x)G(\,dx)
\qquad \text{for all } \varphi,\,\psi\in\Cal S, \tag3.2
$$
where $\,\tilde{}\,$ denotes Fourier transform and $\,\bar{}\,$
complex conjugate. The measure $G$ has
the properties $G(A)=G(-A)$ for all $A\in\Cal B^\nu$, and
$$
\int (1+|x|)^{-r}G(\,dx)<\infty \quad\text{with an appropriate } r>0.
\tag3.3
$$
}

\medskip\noindent
{\it Remark.}\/ The above formulated results are actually not the
Bochner and Bochner--Schwartz theorem in their original form, they
are their consequences. In an Adjustment to Section~3 I formulate
the classical form of these theorems, and explain how the above
formulated results follow from them.

\medskip
The measure $G$ appearing in Theorems~3A and~3B is called the spectral
measure of the stationary field. A measure~$G$ with the same
properties as the measure~$G$ in Theorem~3A or~3B will also be called
a spectral measure. This terminology is justified, since there exists
a stationary random field with spectral measure~$G$ for all such~$G$.

Let us now consider a stationary Gaussian random field (discrete or
generalized one) with spectral measure $G$. We shall denote the space
$L_2([-\pi,\pi)^\nu,\Cal B^\nu,G)$ or $L_2(R^\nu,\Cal B^\nu,G)$ simply
by $L^2_G$. Let $\Cal H_1$ be the real Hilbert space defined by means
of the stationary random field, as it was done in Section~2. Let
$\Cal H^c_1$ denote its complexification, i.e. the elements of
$\Cal H^c_1$ are of the form $X+iY$, \ $X,\,Y\in\Cal H_1$, and the
scalar product is defined as
$(X_1+iY_1,X_2+iY_2)=EX_1X_2+EY_1Y_2+i(EY_1X_2-EX_1Y_2)$.
We are going to construct a unitary transformation $I$ from $L^2_G$
to $\Cal H^c_1$. We shall define the random spectral measure via
this transformation.

Let $\Cal S^c$ denote the Schwartz space of rapidly decreasing, smooth,
complex valued functions with the usual topology of the Schwartz space.
(The elements of $\Cal S^c$ are of the form $\varphi+i\psi$, \
$\varphi,\,\psi\in\Cal S$.)
We make the following observation. The finite linear combinations
$\sum c_ne^{i(n,x)}$ are dense in $L_G^2$ in the discrete field, and
the functions $\varphi\in\Cal S^c$ are dense in $L_G^2$ in the
generalized field case. In the discrete field case this follows from
the Weierstrass approximation theorem, which states that all continuous
functions on $[-\pi,\pi)^\nu$ can be approximated by trigonometrical
polynomials. In the generalized field case let us first observe that
the continuous functions with compact support are dense in $L^2_G$. We
claim that also the functions of the space $\Cal D$ are dense in
$L^2_G$, where $\Cal D$ denotes  the class of (complex valued)
infinitely many times differentiable functions with compact support.
Indeed, if $\varphi\in\Cal D$ is real valued, $\varphi(x)\ge0$ for all
$x\in R^\nu$, $\int\varphi(x)\,dx=1$, we define
$\varphi_t(x)=t^\nu\varphi\(\frac xt\)$, and $f$ is a continuous function
with compact support, then $f*\varphi_t\to f$ uniformly as $t\to\infty$.
Here $*$ denotes convolution. On the other hand, $f*\varphi_t\in\Cal D$
for all $t>0$. Hence $\Cal D\subset\Cal S^c$ is dense in $L^2_G$.

Finally we recall the following result from the theory of distributions.
The mapping $\varphi\to\tilde\varphi$ is an invertible, bicontinuous
transformation from $\Cal S^c$ into $\Cal S^c$. In particular, the
set of functions $\tilde\varphi$, \ $\varphi\in\Cal S$, is also dense
in $L^2_G$.

Now we define the mapping
$$
I\(\sum c_n e^{i(n,x)}\)=\sum c_nX_n \tag3.4
$$
in the discrete case, where the sum is finite, and
$$
I(\widetilde{\varphi+i\psi)}=X(\varphi)+iX(\psi),
\quad \varphi,\,\psi\in\Cal S \tag$3.4'$
$$
in the generalized case.

Obviously,
$$
\align
\left\|\sum c_ne^{i(n,x)}\right\|_{L^2_G}^2
&=\sum\sum c_n\bar c_m\int e^{i(n-m),x}G(\,dx)\\
&=\sum\sum c_n\bar c_m EX_nX_m=E\left|\sum c_nX_n\right|^2,
\endalign
$$
and
$$
\align
&\|\widetilde{\varphi+i\psi}\|^2_{L^2_G}
=\int[\tilde\varphi(x)\bar{\tilde\varphi}(x)
-i\tilde\varphi(x)\bar{\tilde\psi}(x)
+i\tilde\psi(x)\bar{\tilde\varphi}(x)
+\tilde\psi(x)\bar{\tilde\psi}(x)]G(\,dx)\\
&\qquad=EX(\varphi)^2-iEX(\varphi)X(\psi)+iEX(\psi)X(\varphi)
+EX(\psi)^2=E\(|X(\varphi)+iX(\psi)|\)^2.
\endalign
$$
This means that the mapping $I$ from a linear subspace of $L_G^2$
to $\Cal H_1^c$ is norm preserving. Beside this, the subspace where $I$
was defined is dense in $L^2_G$, since the space of continuous functions
is dense in $L^2_G$ if $G$ is a finite measure on the torus
$R^\nu/2\pi\text{\BBB Z}_\nu$, and the space of continuous functions
with a compact support is dense in $L^2_G(R^{\nu})$ if the measure~$G$
satisfies relation~(3.3). Hence the mapping $I$ can be uniquely
extended to a norm preserving transformation from $L^2_G$ to
$\Cal H^c_1$. Since the random variables $X_n$ or $X(\varphi)$
are obtained as the image of some element from $L_G^2$ under this
transformation, $I$ is a unitary transformation from $L^2_G$ to
$\Cal H^c_1$. A unitary transformation preserves not only the
norm, but also the scalar product. Hence
$\int f(x)\bar g(x)G(\,dx)=EI(f)\overline{I(g)}$ for all
$f,\,g\in L^2_G$.

Now we define the random spectral measure $Z_G(A)$ for all
$A\in\Cal B^\nu$ such that $G(A)<\infty$ by the formula
$$
Z_G(A)=I(\chi_A),
$$
where $\chi_A$ denotes the indicator function of the set~$A$. It is clear
that

\medskip
\item{(i)} The random variables  $Z_G(A)$ are complex valued, jointly
Gaussian random variables. (The random variables $\Re Z_G(A)$ and
$\Im G(A)$ with possibly different sets~$A$ are jointly Gaussian.)
\item{(ii)} $EZ_G(A)=0$,
\item{(iii)} $EZ_G(A)\overline {Z_G(B)}=G(A\cap B)$,
\item{(iv)} $\summ_{j=1}^nZ_G(A_j)=Z_G\(\bigcupp_{j=1}^n A_j\)$ if
$A_1,\dots,A_n$ are disjoint sets.

Also the following relation holds.

\item{(v)} $Z_G(A)=\overline{Z_G(-A)}$.

This follows from the relation
\item{(v$'$)} $I(f)=\overline{I(f_-)}$ for all $f\in L^2_G$, where
$f_-(x)=\overline{f(-x)}$.

Relation (v$'$) can be simply checked if $f$ is a finite
trigonometrical polynomial in the discrete field case, or if
$f=\tilde\varphi$, $\varphi\in\Cal S^c$,  in the generalized field
case. (In the case $f=\tilde\varphi$, $\varphi\in\Cal S^c$, the
following argument works. Put
$f(x)=\tilde\varphi_1(x)+i\tilde\varphi_2(x)$ with
$\varphi_1,\varphi_2\in\Cal S$. Then $I(f)=X(\varphi_1)+iX(\varphi_2)$,
and $f_-(x)=\bar{\tilde\varphi}_1(-x)-i\bar{\tilde\varphi}_2(-x)
=\tilde\varphi_1(x)+i(\widetilde{-\varphi_2}(x)$, hence
$I(f_-)=X(\varphi_1)+iX(-\varphi_2)=X(\varphi_1)-iX(\varphi_2)
=\overline{I(f)}$.)
Then a simple limiting procedure implies~(v$'$) in the general
case. Relation~(iii) follows from the identity
$EZ_G(A)\overline {Z_G(B)}=EI(\chi_A)\overline{I(\chi_B)}
=\int \chi_A(x)\overline{\chi_B(x)}G(\,dx)=G(A\cap B)$. The remaining
properties of $Z_G(\cdot)$ are simple consequences of the definition.

\medskip\noindent
{\it Remark.}\/ Property (iv) could have been omitted from the
definition of random spectral measures, since it follows from
property~(iii). To show this it is enough to check that if
$A_1,\dots,A_n$ are disjoint sets, and property~(iii) holds, then
$$
E\(\sum_{j=1}^n Z_G(A_j)-Z_G\(\bigcupp_{j=1}^n A_j\)\)
\overline{\(\sum_{j=1}^n Z_G(A_j)-Z_G\(\bigcupp_{j=1}^n A_j\)\)}=0.
$$

\medskip
Now we introduce the following

\medskip\noindent
{\bf Definition of random spectral measure.} {\it Let $G$ be a spectral
measure. A set of random variables $Z_G(A)$, $G(A)<\infty$, satisfying
(i)--(v) is called a (Gaussian) random spectral measure corresponding
to the spectral measure~$G$.}

\medskip
Given a Gaussian random spectral measure $Z_G$ corresponding to a
spectral measure $G$ we define the stochastic integral
$\int f(x)Z_G(\,dx)$ for an appropriate class of functions~$f$.
Let us first consider simple functions of the form
$f(x)=\sum c_i\chi_{A_i}(x)$, where the sum is finite, and
$G(A_i)<\infty$ for all indices~$i$. In this case we define
$$
\int f(x)Z_G(\,dx)=\sum c_iZ_G(A_i).
$$
Then we have
$$
E\left|\int f(x)Z_G(\,dx)\right|^2=\sum c_i\bar c_jG(A_i\cap A_j)
=\int |f(x)|^2G(\,dx). \tag3.5
$$

Since the simple functions are dense in $L^2_G$, relation~(3.5) enables
us to define $\int f(x)Z_G(\,dx)$ for all $f\in L^2_G$ via
$L_2$-continuity. It can be checked that the expressions
$$
X_n=\int e^{i(n,x)}Z_G(\,dx), \quad n\in\text{\BBB Z}_\nu, \tag3.6
$$
and
$$
X(\varphi)=\int\tilde\varphi(x) Z_G(\,dx), \quad \varphi\in\Cal S,
\tag$3.6'$
$$
defined with the help of the above defined (random) integral and
spectral measure~$Z_G$ are Gaussian stationary random discrete and
generalized field with spectral measure~$G$.

We also have
$$
\int f(x)Z_G(\,dx)=I(f) \quad \text{for all } f\in L_G^2
$$
if we consider the previously defined mapping $I(f)$ with the
stationary random fields defined in~(3.6) and~$(3.6')$. Now we
formulate the following

\medskip\noindent
{\bf Theorem 3.1.} {\it For a stationary Gaussian random field (a
discrete or generalized one) with a spectral measure $G$ there exists
a unique Gaussian random spectral measure $Z_G$ corresponding to the
spectral measure~$G$ on the same probability space as the Gaussian
random field such that relation~(3.6) or~$(3.6')$ holds in the
discrete or generalized field case respectively.

Furthermore
$$
\Cal B(Z_G(A),\; G(A)<\infty)=\left\{
\aligned
&\Cal B(X_n,\;n\in\text{\BBB Z}_\nu) \text{ in the discrete field case,}\\
&\Cal B(X(\varphi),\;\varphi\in\Cal S) \text{ in the generalized field case.}
\endaligned \right. \tag3.7
$$
}

\medskip
We shall say that the random spectral measure $Z_G$ satisfying
Theorem~3.1 together with a Gaussian random field is adapted to
this random field.

\medskip\noindent
{\it Proof of Theorem 3.1.}\/ Given a stationary Gaussian random field
(discrete or stationary one) with a spectral measure~$G$, we have
constructed a random spectral measure $Z_G$ corresponding to the
spectral measure~$G$. Moreover, the random integrals given in
formulas~(3.6) or~$(3.6')$ define the original stationary random
field. Since all random variables $Z_G(A)$ are measurable with
respect to the original random field, relation~(3.6) or~$(3.6')$
implies~(3.7).

To prove the uniqueness, it is enough to observe that because of the
linearity and $L_2$ continuity of stochastic integrals relation~(3.6)
or~$(3.6')$ implies that
$$
Z_G(A)=\int \chi_A(x)Z_G(\,dx)=I(\chi_A)
$$
for a Gaussian random spectral measure corresponding to the spectral
measure~$G$ appearing in Theorem~3.1.

\medskip
Finally we list some additional properties of Gaussian random spectral
measures.

\medskip
\item{(vi)} The random variables $\Re Z_G(A)$ are independent of the
random variables $\Im Z_G(A)$.
\item{(vii)} Random variables of the form $Z_G(A\cup(-A))$ are real
valued. If the sets $A_1\cup(-A_1)$,\dots, $A_n\cup(-A_n)$ are
disjoint, then the random variables $Z_G(A_1)$,\dots, $Z_G(A_n)$ are
independent.
\item{(viii)} If $A\cap(-A)=\emptyset$, then $\Re Z_G(-A)=\Re Z_G(A)$,
$\Im Z_G(-A)=-\Im Z_G(A)$, and the (Gaussian) random variables
$\Re Z_G(A)$ and $\Im Z_G(A)$ are independent with expectation
zero and variance $G(A)/2$.

\medskip
These properties easily follow from (i)--(v). Since $Z_G(\cdot)$
are complex valued Gaussian random variables, to prove the above
formulated independence it is enough to show that the real and imaginary
parts are uncorrelated. We show, as an example, the proof of~(vi).
$$
\align
E\Re Z_G(A)\Im Z_G(B)&=\frac1{4i} E(Z_G(A)+\overline{Z_G(A)})
(Z_G(B)-\overline{Z_G(B)})\\
&=\frac1{4i}E(Z_G(A)+Z_G(-A))(\overline{Z_G(-B)}-\overline{Z_G(B)})\\
&=\frac1{4i}G(A\cap(-B))-\frac1{4i}G(A\cap B)\\
&\qquad+\frac1{4i}G((-A)\cap(-B))-\frac1{4i}G((-A)\cap B)=0
\endalign
$$
for all pairs of sets $A$ and $B$ such that $G(A)<\infty$, $G(B)<\infty$,
since $G(D)=G(-D)$ for all $D\in\Cal B^\nu$. The fact that
$Z_G(A\cup(-A))$ is real valued random variable, and the relations
$\Re Z_G(-A)=\Re Z_G(A)$, $\Im Z_G(-A)=-\Im Z_G(A)$ under the conditions
of~(viii) follow directly from~(v). The remaining statements of~(vii)
and~(viii) can be proved similarly to~(vi) only the calculations are
simpler in this case.

The properties of the random spectral measure $Z_G$ listed above imply
in particular that the spectral measure~$G$ determines the joint
distribution of the corresponding random variables $Z_G(B)$,
$B\in\Cal B^\nu$.

\medskip\noindent
{\bf Attachment to Section~3.} {\it A more detailed discussion about
the spectral representation of the covariance function of stationary
random fields.}

\medskip\noindent
The results formulated under the name of Bochner and Bochner--Schwartz
theorem (I write this, because actually I presented not these theorems
but an important consequence of them) have the following content.
Given a finite, even measure~$G$ on the torus
$R^{\nu}/2\pi\text{\BBB Z}_\nu$ one can define a (Gaussian)
discrete stationary field with correlation function satisfying~(3.1)
with this measure~$G$. For an even measure $G$ on~$R^\nu$
satisfying~(3.3) there exists a (Gaussian) generalized stationary
field with correlation function defined in formula~(3.2) with this
measure $G$. The Bochner and Bochner--Schwartz theorems state that the
correlation function of all (Gaussian) discrete stationary fields,
respectively of all stationary generalized fields can be represented
in such a way. Let us explain this in more detail.

First I formulate the following

\medskip\noindent
{\bf Proposition~3C.} {\it Let $G$ be a finite measure on the torus
$R^{\nu}/2\pi\text{\BBB Z}_\nu$ such that $G(A)=G(-A)$ for all measurable
sets~$A$. Then there exists a Gaussian discrete stationary field $X_n$,
$n\in\text{\BBB Z}_\nu$, with expectation zero such that its correlation
function $r(n)=EX_kX_{k+n}$, $n,k\in\text{\BBB Z}_\nu$, is given by
formula~(3.1) with this measure~$G$.

Let $G$ be a measure on~$R^\nu$ satisfying~(3.3) and such that
$G(A)=G(-A)$ for all measurable sets~$A$. Then there exists a Gaussian
stationary generalized field $X(\varphi)$, $\varphi\in\Cal S$, with
expectation $EX(\varphi)=0$ for all $\varphi\in\Cal S$ such that
its covariance function $EX(\varphi)X(\psi)$, $\varphi,\psi\in\Cal S$,
satisfies formula~(3.2) with this measure~$G$.

Moreover, the correlation function $r(n)$ or $EX(\varphi)X(\psi)$,
$\varphi,\psi\in\Cal S$, determines the measure~$G$ uniquely.}

\medskip\noindent
{\it Proof of Proposition~3C.}\/ By Kolmogorov's theorem about
the existence of random processes with consistent finite dimensional
distributions it is enough to prove the following statement to show
the existence of the Gaussian discrete stationary field with the
demanded properties. For any points
$n_1,\dots,n_p\in\text{\BBB Z}_\nu$ there exists a Gaussian random
vector $(X_{n_1},\dots,X_{n_p})$ with expectation zero and covariance
matrix $EX_{n_j}X_{n_k}=r(n_j-n_k)$. (Observe that the function~$r(n)$
is real valued, $r(n)=r(-n)$, because of the evenness of the spectral
measure~$G$.) Hence it is enough to check that the corresponding
matrix is positive definite, i.e. $\summ_{j,k} c_jc_k r(n_j-n_k)\ge0$
for all real vectors $(c_1,\dots,c_p)$. This relation holds, because
$\summ_{j,k} c_jc_k r(n_j-n_k)=\int
|\summ_j c_je^{i(n_j,x)}|^2\,G(\,dx)\ge0$ by formula~(3.1).

It can be proved similarly that in the generalized field
case there exists a Gaussian random field with expectation zero whose
covariance function satisfies formula~(3.2). (Let us observe that
the relation $G(A)=G(-A)$ implies that $EX(\varphi)X(\psi)$ is a real
number for all $\varphi,\,\psi\in\Cal S$, since
$EX(\varphi)X(\psi)=\overline{EX(\varphi)X(\psi)}$ in this case.
In the proof of this identity we exploit that
$\bar{\tilde f}(x)=\tilde f(-x)$ for a real valued function~$f$.)
We also have to show that a random field with such a distribution is
a generalized field, i.e. it satisfies properties~a) and~b) given in
the definition of generalized fields. It is not difficult to show
that if $\varphi_n\to\varphi$ in the topology of the space $\Cal S$,
then $E[X(\varphi_n)-X(\varphi)]^2
=\int|\tilde\varphi_n(x)-\tilde\varphi(x)|^2 G(\,dx)\to0$ as
$n\to\infty$, hence property~b) holds. (Here we exploit that the
transformation $\varphi\to\tilde\varphi$ is bicontinuous in the
space~$\Cal S$.) Property~a) also holds, because, as it is not
difficult to check with the help of formula~(3.2),
$$
\align
&E[a_1X(\varphi_1)+a_2X(\varphi_2)
-X(\varphi(a_1\varphi_1+a_2\varphi_2)]^2\\
&\qquad =\int\left|a_1\tilde\varphi_1(x)+a_2\tilde\varphi_2(x)
-(\widetilde{a_1\varphi_1+a_2\varphi_2})(x)\right|^2G(\,dx)=0.
\endalign
$$
It is clear that the Gaussian random field constructed in such a way
is stationary.
 
Finally, as we have seen in our considerations in the main text, the
correlation function determines the integral $\int f(x)\,G(\,dx)$
for all continuous functions~$f$ with a bounded support, hence it
also determines the measure~$G$.

\medskip
The Bochner and Bochner--Schwartz theorems enable us to show that the
correlation function of all stationary (Gaussian) fields (discrete or
generalized one) can be presented in the above way with an appropriate
spectral measure~$G$. To see this let us formulate these results in
their original form.

To formulate Bochner's theorem first we have to introduce the following
notion.

\medskip\noindent
{\bf Definition of positive definite functions.} {\it Let $f(x)$ be a
(complex valued) function on $\text{\BBB Z}_\nu$ (or on $R^\nu$). We
say that  $f(\cdot)$ is a positive definite function if for all
parameters~$p$, complex numbers $c_1,\dots,c_p$ and points
$x_1,\dots,x_p$ in  $\text{\BBB Z}_\nu$ (or in $R^\nu$) the inequality
$$
\sum_{j=1}^p\sum_{k=1}^p c_j\bar c_k f(x_j-x_k)\ge0
$$
holds.}

\medskip
A simple example for positive definite functions is the function
$f(x)=e^{i(t,x)}$, where $t\in\text{\BBB Z}_\nu$ in the discrete, and
$t\in R^\nu$ in the continuous case. Bochner's theorem provides a
complete description of positive definite functions.

\medskip\noindent
{\bf Bochner's theorem. (Its original form)} {\it A complex valued
function $f(x)$ defined on $\text{\BBB Z}_\nu$ is positive definite
if and only if it can be written in the form
$f(x)=\int e^{i(t,x)}G(\,dx)$ for all
$x\in\text{\BBB Z}_\nu$ with a finite measure~$G$ on the
torus~$R^\nu/2\pi\text{\BBB Z}_\nu$. The measure~$G$ is uniquely
determined.

A complex valued function $f(x)$ defined on $R^\nu$ is continuous and
positive definite if and only if it can be written in the form
$f(x)=\int e^{i(t,x)}G(\,dx)$ for all $x\in R^\nu$ with a finite
measure~$G$ on $R^\nu$. The measure~$G$ is uniquely determined.}

\medskip
It is not difficult to see that the covariance function $r(n)=EX_kX_{k+n}$,
($EX_n=0$), $k,n\in\text{\BBB Z}_\nu$, of a stationary (Gaussian)
random field~$X_n$ is a positive definite
function, since
$\summ_{j,k} c_j\bar c_kr(n_j-n_k)=E|\summ_j c_jX_{n_j}|^2>0$
for any vector $(c_1,\dots,c_p)$. Hence Bochner's theorem can be applied
for it. Beside this, the relation $r(n)=r(-n)$ together with the
uniqueness of the measure~$G$ appearing in Bochner's theorem imply that
the identity $G(A)=G(-A)$ holds for all measurable sets~$G$. This
implies the result formulated in the main text under the name Bochner's
theorem.

\medskip
The Bochner--Schwartz theorem formulates an analogous representation
of positive definite generalized functions in~$\Cal S'$ as the
Fourier transforms of positive generalized functions in~$\Cal S'$
together with an analogous result about generalized functions in the
space~$\Cal D'$. To formulate it we have to introduce some definitions.
First we have to clarify what a positive generalized function means.
We introduce this notion both in the space~$\Cal S'$ and $\Cal D'$,
and then we characterize them in a Theorem.

\medskip\noindent
{\bf Definition of positive generalized functions.} {\it A linear
functional $F\in\Cal S'$ (or $F\in\Cal D'$) is called a positive
definite generalized function if for all such $\varphi\in\Cal S$
(or $\varphi\in\Cal D$) test functions for which $\varphi(x)\ge0$ for
all $x\in R^\nu$ $(F,\varphi)\ge0$.}

\medskip\noindent
{\bf Theorem about the representation of positive generalized
functions.} {\it All positive generalized functions $F\in\Cal S'$ can
be given in the form $(F,\varphi)=\int \varphi(x)\mu(\,dx)$,
where $\mu$ is a polynomially increasing measure on $R^\nu$, i.e.\
it satisfies the relation $\int(1+|x|^2)^{-p}\mu(\,dx)<\infty$ with
some $p>0$. Similarly, all positive generalized functions in
$\Cal D'$ can be given in the form
$(F,\varphi)=\int \varphi(x)\mu(\,dx)$ with such a measure $\mu$
on $R^\nu$ which is finite in all bounded regions. The generalized
function~$F$ uniquely determines the measure~$\mu$ in both cases.}

\medskip
We also need the introduction of a technical notion and a result
related to it. Let us remark that if $\varphi\in\Cal S^c$ and
$\psi\in\Cal S^c$, then also their product $\varphi\psi\in\Cal S^c$.
The analogous result also holds in the space $\Cal D$.

\medskip\noindent
{\bf Definition of multiplicatively positive generalized functions.}
{\it A generalized function $\Cal F\in\Cal S'$ (or $F\in\Cal D'$) is
multiplicatively positive if
$(F,\varphi\bar\varphi)=(F,|\varphi|^2)\ge0$ for all
$\varphi\in\Cal S^c$ (or in $\varphi\in\Cal D$).}

\medskip\noindent
{\bf Theorem about the characterization of multiplicatively positive
generalized functions.} {\it A generalized function $F\in\Cal S'$
(or $F\in\Cal D'$) is multiplicatively positive if and only if it is
positive.}

\medskip
Now I introduce the definition of positive definite generalized
functions.

\medskip\noindent
{\bf Definition of positive definite generalized functions.} {A
generalized function $F\in\Cal S'$ (or $F\in\Cal D'$) is positive
definite if $(F,\varphi*\varphi^*)\ge0$ for all $\varphi\in\Cal S^c$
(of $\varphi\in\Cal D$), where $\varphi^*(x)=\overline {\varphi(-x)}$,
and $*$ denotes convolution, i.e. $\varphi*\varphi^*(x)=\int\varphi(t)
\overline{\varphi(t-x)}\,dt$.}

\medskip
We refer to~[15] for an explanation why this definition of positive
definite generalized functions is natural. Let us remark that if
$\varphi,\psi\in\Cal S^c$, then $\varphi*\psi\in\Cal S^c$, and the
analogous result holds in~$\Cal D$. The original version of
the Bochner--Schwartz theorem has the following form.

\medskip\noindent
{\bf Bochner--Schwartz theorem. (Original form)} {\it Let $F$ be a
positive definite generalized function in the space $\Cal S'$ (or
$\Cal D'$). Then it is the Fourier transform of a polynomially
increasing measure~$\mu$ on $R^\nu$, i.e. the identity
$(F,\varphi)=\int\tilde\varphi(x)\,\mu(\,dx)$ holds for all
$\varphi\in\Cal S^c$ (or $\varphi\in\Cal D$) with a measure $\mu$
that satisfies the relation $\int(1+|x|^2)^{-p}\mu(\,dx)<\infty$
with an appropriate $p>0$. The generalized function~$F$ uniquely
determines the measure~$\mu$. On the other hand, if $\mu$ is a
polynomially increasing measure on $R^\nu$, then the formula
$(F,\varphi)=\int \tilde\varphi(x)\mu(\,dx)$ with $\varphi\in\Cal S^c$
(or $\varphi\in\Cal D$) defines a positive definite generalized
function~$F$ in the space $\Cal S'$ (or $\Cal D'$).}

\medskip\noindent
{\it Remark.} It is a remarkable and surprising fact that the class
of positive definite generalized functions are represented by the
same class of measures~$\mu$ in the spaces $\Cal S'$ and $\Cal D'$.
(In the representation of positive generalized functions the class
of measures~$\mu$ considered in the case of $\Cal D'$ is much
larger, than in the case of $\Cal S'$.) Let us remark that in the
representation of the positive definite generalized functions in
$\Cal D'$ the function $\tilde\varphi$ we integrate is not in the
class~$\Cal D$, but in the space~$\Cal Z$ consisting of the Fourier
transforms of the functions in~$\Cal D$.

\medskip
It is relatively simple to prove the representation of positive
definite generalized functions given in the Bochner--Schwartz
theorem for the class~$\Cal S'$. Some calculation shows that if $F$ is
a positive definite generalized function, then its Fourier transform
is a multiplicatively positive generalized function. Indeed, since
the Fourier transform of the convolution $\varphi*\psi(x)$ equals
$\tilde\varphi(t)\tilde\psi(t)$, and the Fourier transform of
$\varphi^*(x)=\overline{\varphi(-x)}$ equals $\overline{\tilde\varphi(t)}$,
the Fourier transform of $\varphi*\varphi^*(x)$ equals
$\tilde\varphi(t)\bar{\tilde\varphi}(t)$. Hence the positive
definitiveness property of the generalized function~$F$ and the
definition of the Fourier transform of generalized functions imply that
$(\tilde F,\tilde\varphi\bar{\tilde\varphi})
=(2\pi)^{\nu}(F,\varphi*\varphi^*)\ge0$ for all $\varphi\in\Cal S^c$.
Since every function of~$\Cal S^c$ is the Fourier transform
$\tilde\varphi$ of some function $\varphi\in\Cal S^c$ this implies
that $\tilde F$ is a multiplicatively positive and as a consequence
a positive generalized function in $\Cal S'$. Such generalized
functions have a good representation with the help of a polynomially
increasing positive measure~$\mu$. Since
$(F,\varphi)=(2\pi)^{-\nu}(\tilde F,\tilde\varphi)$ it is not difficult
to prove the Bochner--Schwartz theorem for the space~$\Cal S'$ with the
help of this fact. The proof is much harder if the space~$\Cal D'$
is considered, but we do not need that result.

The Bochner--Schwartz theorem in itself is not sufficient to describe
the correlation function of a generalized random fields. We still
need another important result of Laurent Schwartz which gives useful
information about the behaviour of (Hermitian) bilinear functionals
in $\Cal S^c$ and some additional information about the behaviour of
translation invariant (Hermitian) bilinear functionals in this space.
To formulate these results first we introduce the following definition.

\medskip\noindent
{\bf Definition of Hermitian bilinear and translation invariant
Hermitian bilinear functionals in the space $\Cal S^c$.} {\it A function
$B(\varphi,\psi)$, $\varphi,\psi\in\Cal S^c$, is a Hermitian bilinear
functional in the space $\Cal S^c$ if for all fixed $\psi\in\Cal S^c$
$B(\varphi,\psi)$ is a continuous linear functional of the
variable~$\psi$ in the topology of~$\Cal S^c$, and for all fixed
$\varphi\in\Cal S^c$ $\overline{B(\varphi,\psi)}$ is a continuous
linear functional of the variable~$\psi$ in the topology of~$\Cal S^c$.

A Hermitian bilinear functional $B(\varphi,\psi)$ in $\Cal S^c$ is
translation invariant if it does not change by a simultaneous shift
of its variables $\varphi$ and $\psi$, i.e.\ if
$B(\varphi(x),\psi(x))=B(\varphi(x+h),\psi(x+h))$ for all $h\in R^\nu$.}

\medskip\noindent
{\bf Definition of positive definite Hermitian bilinear functionals.}
{\it We say that a Hermitian bilinear functional $B(\varphi,\psi)$ in
$\Cal S^c$ is positive definite if $B(\varphi,\varphi)\ge0$ for all
$\varphi\in\Cal S^c$.}

\medskip
The next result characterizes the Hermitian bilinear and translation
invariant Hermitian bilinear functionals in~$\Cal S^c$.

\medskip\noindent
{\bf Theorem 3D.} {\it All Hermitian bilinear functionals
$B(\varphi,\psi)$ in $\Cal S^c$ can be given in the form
$B(\varphi,\psi)=(F_1,\varphi(x)\overline{\psi(y)})$,
$\varphi,\psi\in\Cal S^c$, where $F_1$ is a continuous linear
functional on $\Cal S^c\times\Cal S^c$, i.e. it is a generalized
function in~${\Cal S_{2\nu}}'$.

A translation invariant Hermitian bilinear functional in $\Cal S^c$
can be given in the form $\Cal B(\varphi,\psi)=(F,\varphi*\psi^*)$,
$\varphi,\psi\in\Cal S^c$, where $F\in\Cal S,$,
$\psi^*(x)=\overline\psi(-x)$, and $*$ denotes convolution.

The Hermitian bilinear form $B(\varphi,\psi)$ determines the generalized
functions $F_1$, and if it is translation invariant then also the
generalized function $F$ uniquely. Beside this, for  all functionals
$F_1\in{\Cal S^{2\nu}}'$ and $F\in\Cal S'$ the above formulas define
a Hermitian bilinear functional and a translation invariant Hermitian
bilinear functional in~$\Cal S^c$ respectively.}

\medskip
Let us consider a Gaussian generalized random field $X(\varphi)$,
$\varphi\in\Cal S$, with expectation zero together with its
correlation function $B(\varphi,\psi)=EX\varphi)X(\psi)$, \
$\varphi,\psi\in\Cal S$. More precisely, let us consider the
complexification $X(\varphi_1+i\varphi_2)=X(\varphi_1)+iX(\varphi_2)$
of this random field and its correlation function
$B(\varphi,\psi)=EX(\varphi)\overline{X(\psi)}$,
$\varphi,\psi\in\Cal S^c$. This correlation function
$B(\varphi,\psi)$ is a translation invariant Hermitian bilinear
functional in $\Cal S^c$, hence it can be written in the form
$B(\varphi,\psi)=(F,\varphi*\psi^*)$ with an appropriate
$F\in\Cal S'$. Moreover, $B(\varphi,\varphi)\ge0$ for all
$\varphi\in\Cal S^c$, and this means that the generalized
function $F\in\Cal S'$ corresponding to $B(\varphi,\psi)$ is
positive definite. Hence the Bochner--Schwartz theorem can be applied
for it, and it yields that
$$
EX(\varphi)X(\psi)=\int \widetilde{\varphi*\psi^*}(x)\,G(\,dx)
=\int \tilde\varphi(x)\bar{\tilde\psi}(x)\,G(\,dx) \quad\text{for all }
\varphi,\,\psi\in \Cal S^c
$$
with a uniquely determined, polynomially increasing measure~$G$
on~$R^\nu$. To prove Theorem~3B we still have to show that $G$ is an
even measure. In the proof of this statement we exploit that for a
real valued function $\varphi\in\Cal S$ the random variable
$X(\varphi)$ is also real valued. Hence if $\varphi,\psi\in\Cal S$,
then $EX(\varphi)X(\psi)=\overline{EX(\varphi)X(\psi)}$. Beside this
$\tilde\varphi(-x)=\bar{\tilde\varphi}(x)$ and
$\tilde\psi(-x)=\bar{\tilde\psi}(x)$ in this case. Hence
$$
\align
\int \tilde\varphi(x)\bar{\tilde\psi}(x)\,G(\,dx)
&=\int \bar{\tilde\varphi}(x)\tilde\psi(x)\,G(\,dx)\\
&=\int \tilde\varphi(-x)\bar{\tilde\psi}(-x)\,G(\,dx)
=\int \tilde\varphi(x)\bar{\tilde\psi}(x)\,G^-(\,dx)
\endalign
$$
for all $\varphi,\psi\in\Cal S$, where $G^-(A)=G(-A)$ for all
$A\in\Cal B^\nu$. This relation implies that the measures~$G$
and~$G^-$ agree. The proof of Theorem~3B is completed.

\beginsection 4. Multiple Wiener--It\^o integrals.

In this section we define the so-called multiple Wiener--It\^o
integrals, and we prove their most important properties with the
help of It\^o's formula, whose proof is postponed to the next
section. More precisely, we discuss in this section a modified
version of the Wiener--It\^o integrals with respect to a random
spectral measure rather than with respect to a random measure with
independent increments. This modification makes it necessary to
slightly change the definition of the integral. This modified
Wiener--It\^o integral seems to be a more useful tool than the
original one or the Wick polynomials, because it enables us to
describe the action of shift transformations.

Let $G$ be the spectral measure of a stationary Gaussian field
(discrete or generalized one). We define the following
{\it real}\/ Hilbert spaces $\bar{\Cal H}_G^n$ and $\Cal H_G^n$,
$n=1,2,\dots$. We have $f_n\in\bar{\Cal H}_G^n$ if and only if
$f_n=f_n(x_1,\dots,x_n)$, \ $x_j\in R^\nu$, $j=1,2,\dots,n$, is a
complex valued function of $n$ variables, and

\medskip
\item{(a)} $f_n(-x_1,\dots,-x_n)=\overline{f_n(x_1,\dots,x_n)}$,
\item{(b)}
$\|f_n\|^2=\int|f_n(x_1,\dots,x_n)|^2G(\,dx_1)\dots G(\,dx_n)<\infty$.

\medskip
Relation~(b) also defines the norm in $\bar{\Cal H}^n_G$. The
subspace $\Cal H^n_G\subset\bar{\Cal H}_G^n$ contains those functions
$f_n\in\bar{\Cal H}_G^n$ which are invariant under permutations of
their arguments, i.e.

\medskip
\item{(c)} $f_n(x_{\pi(1)},\dots,x_{\pi(n)}))=f_n(x_1,\dots,x_n)$
for all $\pi\in\Pi_n$, where $\Pi_n$ denotes the group of all
permutations of the set $\{1,2,\dots,n\}$.

The norm in $\Cal H_G^n$ is defined in the same way as in
$\bar{\Cal H}_G^n$. Moreover, the scalar product is also similarly
defined, namely if $f,\,g\in\bar{\Cal H}_G^n$, then
$$
\align
(f,g)&=\int f(x_1,\dots,x_n)\overline{g(x_1,\dots,x_n)}
G(\,dx_1)\dots G(\,dx_n)\\
&=\int f(x_1,\dots,x_n)g(-x_1,\dots,-x_n)G(\,dx_1)\dots G(\,dx_n).
\endalign
$$
Because of the symmetry $G(A)=G(-A)$ of the spectral measure
$(f,g)=\overline{(f,g)}$, i.e. the scalar product $(f,g)$ is a real
number for all $f,\,g\in\bar{\Cal H}_G^n$. This means that
$\bar{\Cal H}_G^n$ is a real Hilbert space.
We also define $\Cal H_G^0=\bar{\Cal H}_G^0$ as
the space of real constants with the norm $\|c\|=|c|$.  We remark
that $\bar{\Cal H}_G^n$ is actually the $n$-fold direct product of
$\Cal H_G^1$, while $\Cal H_G^n$ is the $n$-fold symmetrical direct
product of $\Cal H^1_G$. Condition~(a) means heuristically that
$f_n$ is the Fourier transform of a real valued function.

Finally we define the so-called Fock space $\text{Exp\,}\Cal H_G$
whose elements
are sequences of functions $f=(f_0,f_1,\dots)$, $f_n\in\Cal H_G^n$
for all $n=0,1,2,\dots$, such that
$$
\|f\|^2=\sum_{n=0}^\infty \frac1{n!}\|f_n\|^2<\infty.
$$
Given a function $f\in\bar{\Cal H}^n_G$ we define $\Sym f$ as
$$
\Sym f(x_1,\dots,x_n)=\frac1{n!}\sum_{\pi\in\Pi_n}
f(x_{\pi(1)},\dots,x_{\pi(n)}).
$$
Clearly, $\Sym f\in\Cal H_G^n$, and
$$
\|\Sym f\|\le \|f\|. \tag4.1
$$

Let $Z_G$ be a Gaussian random spectral measure corresponding to the
spectral measure~$G$ on a probability space $(\Omega,\Cal A,P)$. We
shall define the $n$-fold Wiener--It\^o integrals
$$
I_G(f_n)=\frac1{n!}\int f_n(x_1,\dots,x_n)Z_G(\,dx_1)\dots Z_G(\,dx_n)
\quad \text{for } f_n\in\bar{\Cal H}_G^n
$$
and
$$
I_G(f)=\sum_{n=0}^\infty I_G(f_n)\quad \text{for }
f=(f_0,f_1,\dots)\in\text{Exp}\,\Cal H_G.
$$
We shall see that  $I_G(f_n)=I_G(\Sym f_n)$ for all
$f_n\in\bar{\Cal H}_G^n$. Therefore, it would have been sufficient
to define the Wiener--It\^o integral only for functions in
$\Cal H_G^n$. Nevertheless, some arguments become simpler if we work
in $\bar{\Cal H}_G^n$. In the definition of Wiener--It\^o integrals
first we restrict ourselves to the case when the spectral measure is
non-atomic, i.e. $G(\{x\})=0$ for all $x\in R^\nu$. This condition is
satisfied in all interesting cases. However, we shall later show how
one can get rid of this restriction.

First we define a subclass
$\hat{\bar{\Cal H}}_G^n\subset\bar{\Cal H}_G^n$ of simple functions,
and define the Wiener--It\^o integrals for the functions of this
subclass.

Let $\Cal D=\{\Delta_j,\;j=\pm1,\pm2,\dots,\pm N\}$ be a finite
collection of bounded, measurable sets in $R^\nu$  indexed by the
integers $\pm1,\dots,\pm N$. We say that $\Cal D$ is a regular
system if
$\Delta_j=-\Delta_{-j}$, and $\Delta_j\cap\Delta_l=\emptyset$
if $j\neq l$ for all $j,l=\pm1,\pm2,\dots,\pm N$. A function
$f\in\bar{\Cal H}_G^n$ is adapted to this system $\Cal D$ if
$f(x_1,\dots,x_n)$ is constant on the sets
$\Delta_{j_1}\times\Delta_{j_2}\times\cdots\times\Delta_{j_n}$, \
$j_l=\pm1,\dots,\pm N$, $l=1,2,\dots,n$,  it vanishes outside these
sets and also on the sets for which $j_l=\pm j_{l'}$ for some
$l\neq l'$. A function $f\in\bar{\Cal H}_G^n$ is in the class
$\hat{\bar{\Cal H}}_G^n$ of simple functions if it is adapted to
some regular systems
$\Cal D=\{\Delta_j,\;j\pm1,\dots,\pm N\}$, and its Wiener--It\^o
integral with respect to $Z_G$ is defined as
$$
\aligned
&\int f(x_1,\dots,x_n)Z_G(\,dx_1)\dots Z_G(\,dx_n) \\
&\qquad =n!I_G(f)=\sum\Sb j_l=\pm1,\dots,\pm N\\l=1,2,\dots,n\endSb
f(x_{j_1},\dots,x_{j_n})Z_G(\Delta_{j_1})\cdots Z_G(\Delta_{j_n}),
\endaligned \tag4.2
$$
where $x_j\in\Delta_j$, $j=\pm1,\dots,\pm N$. We remark that although
the regular system $\Cal D$ to which $f$ is adapted, is not uniquely
determined (the elements of $\Cal D$ can be divided to smaller
sets), the integral defined in~(4.2) is meaningful, i.e.
it does not depend on the choice of $\Cal D$. This can be seen by
observing that a refinement of a regular system $\Cal D$ adapted to
the function $f$ yields the same value for the sum defining $n!I_G(f)$ in
formula~(4.2) as the original one. This follows from the additivity
of the random spectral measure $Z_G$ formulated in its property~(iv),
since this implies that each term
$f(x_{j_1},\dots,x_{j_n})Z_G(\Delta_{j_1})\cdots Z_G(\Delta_{j_n})$
in the sum at the right-hand side of formula (4.2) corresponding to
the original regular
system equals the sum of all such terms $f(x_{j_1},\dots,x_{j_n})
Z_G(\Delta'_{j'_1})\cdots Z_G(\Delta'_{j'_n})$ in the sum
corresponding to the refined partition for which
$\Delta'_{j'_1}\times\cdots\times\Delta'_{j'_n}\subset
\Delta_{j_1}\times\cdots\times\Delta_{j_n}$.

By property~(vii) of the random spectral measures all products
$Z_G(\Delta_{j_1})\cdots Z_G(\Delta_{j_n})$ with non-zero
coefficient in~(4.2) are products of independent random variables.
We had this property in mind when requiring the condition that the
function $f$ vanishes on a product
$\Delta_{j_1}\times\cdots\times\Delta_{j_n}$ if $j_l=\pm j_{l'}$ for
some $l\neq l'$. This condition is interpreted in the literature
as discarding the hyperplanes $x_l=x_{l'}$ and $x_l=-x_{l'}$,
\ $l,l'=1,2,\dots,n$, $l\neq l'$, from the domain of integration.
Property~(a) of the functions in $\bar{\Cal H}_G^n$ and
property~(v) of the random spectral measures imply that
$I_G(f)=\overline{I_G(f)}$, i.e. $I_G(f)$ is a real valued random
variable for all $f\in\hat{\bar{\Cal H}}_G^n$. The relation
$$
EI_G(f)=0, \quad \text{for }f\in\hat{\bar{\Cal H}}_G^n,
\quad n=1,2,\dots \tag4.3
$$
also holds. Let $\hat{\Cal H}_G^n=\Cal H_G^n\cap\hat{\bar{\Cal H}}_G^n$.
If $f\in\hat{\bar{\Cal H}}_G^n$, then  $\Sym f\in\hat{\Cal H}_G^n$, and
$$
I_G(f)=I_G(\Sym f). \tag4.4
$$
Relation~(4.4) follows immediately from the observation that
$Z_G(\Delta_{j_1})\cdots Z_G(\Delta_{j_n})=Z_G(\Delta_{\pi(j_1)})\cdots
Z_G(\Delta_{\pi(j_n)})$ for all $\pi\in\Pi_n$.
We also claim that
$$
EI_G(f)^2\le\frac1{n!}\|f\|^2 \quad{\text for \ }
f\in\hat{\bar{\Cal H}}_G^n, \tag4.5
$$
and
$$
EI_G(f)^2=\frac1{n!}\|f\|^2 \quad{\text for \ } f\in\hat{\Cal H}_G^n.
\tag$4.5'$
$$
Because of (4.1) and (4.4) it is enough to check~$(4.5')$.

Let $\Cal D$ be a regular system of sets in $R^\nu$,
$j_1,\dots,j_n$ and $k_1,\dots,k_n$ be indices such that
$j_l\neq\pm j_{l'}$, $k_l\neq\pm k_{l'}$ if $l\neq l'$. Then
$$
\align
&EZ_G(\Delta_{j_1})\cdots Z_G(\Delta_{j_n})
\overline{Z_G(\Delta_{k_1})\cdots Z_G(\Delta_{k_n})}\\
&\qquad\qquad=\left\{
\aligned
&G(\Delta_{j_1})\cdots G(\Delta_{j_n}) \quad{\text if \ }
\{j_1,\dots,j_n\}=\{k_1,\dots,k_n\}, \\
&0 \quad \text{otherwise.}
\endaligned \right.
\endalign
$$

To see the last relation one has to observe that the product on the
left-hand side can be written as a product of independent random
variables because of property~(vii) of the random spectral measures.
If $\{j_1,\dots,j_n\}\neq\{k_1,\dots,k_n\}$, then there is an index~$l$
such that either $j_l\neq\pm k_{l'}$ for all $1\le l'\le n$, or there
exists an index $l'$, $1\le l'\le n$, such that $j_l=-k_{l'}$. In the
first case $Z_G(\Delta_{j_l})$ is independent of the remaining
coordinates of the vector
$(Z_G(\Delta_{j_1}),\dots,Z_G(\Delta_{j_n}),
\overline{Z_G(\Delta_{k_1})},\dots,\overline{Z_G(\Delta_{k_n})})$,
and $EZ_G(\Delta_{j_l})=0$. Hence the expectation of the investigated
product equals zero, as we claimed. If ${j_l}=-k_{l'}$ with some index
$l'$, then a different argument is needed, since $Z_G(\Delta_{j_l})$
and $Z_G(-\Delta_{j_l})$ are not independent. In this case we can
state that since $j_p\neq\pm j_l$ if $p\neq l$, and
$k_q\neq\pm j_l$ if $q\neq l'$, the vector
$(Z_G(\Delta_{j_l}),Z_G(-\Delta_{j_l}))$ is independent of the
remaining coordinates of the above random vector. On the other hand,
the product $Z_G(\Delta_{j_l})\overline{Z_G(-\Delta_{j_l}})$
has zero expectation, since
$EZ_G(\Delta_{j_l})\overline{Z_G(-\Delta_{j_l})}
=G(\Delta_{j_l}\cap(-\Delta_{j_l}))=0$ by property~(iii) of the
random spectral measures and the relation
$\Delta_{j_l}\cap(-\Delta_{j_l})=\emptyset$. Hence the expectation
of the considered product equals zero also in this case. If
$\{j_1,\dots,j_n\}=\{k_1,\dots,k_n\}$, then
$$
EZ_G(\Delta_{j_1})\cdots Z_G(\Delta_{j_n})
\overline{Z_G(\Delta_{k_1})\cdots Z_G(\Delta_{k_n})}
=\prodd_{l=1}^n EZ_G(\Delta_{j_l})\overline{Z_G(\Delta_{j_l})}
=\prodd_{l=1}^n G(\Delta_{j_l}).
$$

Therefore for a function $f\in\hat{\Cal H}_G^n$
$$
\align
EI_G(f)^2&=\(\frac1n\)^2\sum\sum f(x_{j_1},\dots,x_{j_n})
\overline{f(x_{k_1},\dots,x_{k_n})} \\
&\qquad\qquad\qquad EZ_G(\Delta_{j_1})\cdots Z_G(\Delta_{j_n})
\overline{Z_G(\Delta_{k_1})\cdots Z_G(\Delta_{k_n})}\\
&=\(\frac1{n!}\)^2\sum |f(x_{j_1},\dots,x_{j_n})|^2
G(\Delta_{j_1})\cdots G(\Delta_{j_n}) \cdot n! \\
&=\frac1{n!}\int |f(x_1,\dots,x_n)|^2G(\,dx_1)\cdots G(\,dx_n)
=\frac1{n!}\|f\|^2.
\endalign
$$

We claim that Wiener--It\^o integrals of different order are
uncorrelated. More explicitly, take two functions
$f\in\hat{\bar{\Cal H}}^n_G$ and $f'\in\hat{\bar{\Cal H}}^{n'}_G$
such that $n\neq n'$. Then we have
$$
EI_G(f)I_G(f')=0 \quad \text{if \ }f\in \hat{\bar{\Cal H}}^n_G, \;\;
f'\in\hat{\bar{\Cal H}}^{n'}_G, \text{ and \ } n\neq n'. \tag4.6
$$
To see this relation observe that a regular system $\Cal D$ can be
chosen is such a way that both $f$ and $f'$ are adapted to it.
Then a similar, but simpler argument as the previous one shows that
$$
EZ_G(\Delta_{j_1})\cdots Z_G(\Delta_{j_n})
\overline{Z_G(\Delta_{k_1})\cdots Z_G(\Delta_{k_{n'}})}=0
$$
for all sets of indices $\{j_1,\dots,j_n\}$ and $\{k_1,\dots,k_{n'}\}$
if $n\neq n'$, hence the sum expressing $EI_G(f)I_G(f')$ in this case
equals zero.

\medskip
We show that $\hat{\bar{\Cal H}}_G^n$ is dense in $\bar{\Cal H}_G^n$
(and $\hat{\Cal H}_G^n$ is dense in $\Cal H_G^n$). First we show that
this property can be reduced to Statement~A formulated below.
In Statement~A we reduce the statement about the good approximability
of a general function $f\in\bar{\Cal H}_G^n$ to the good
approximability of the indicator function $\chi_A$ of a bounded set
$A\in\Cal B^{n\nu}$ such that $A=-A$ by a
function of the form $g=\chi_B\in\hat{\bar{\Cal H}}_G^n$. (Observe
that $\chi_A\in\bar{\Cal H}_G^n$ for a bounded set $A\in\Cal B^{n\nu}$
if and only if $A=-A$.) However, we have to formulate Statement~A
in a more complicated form, because only in such a way can we reduce
the statement about the good approximability of a general, possibly
complex valued function~$f\in\bar{\Cal H}_G^n$ by a function in
$g\in\hat{\bar{\Cal H}}_G^n$ to Statement~A.

\medskip\noindent
{\it Statement A.}\/ Let $A\in\Cal B^{n\nu}$ be a bounded, symmetric
set, i.e. let $A=-A$. Then for any $\e>0$ there is a function
$g\in\hat{\bar{\Cal H}}_G^n$ such that $g=\chi_B$ with some set
$B\in\Cal B^{n\nu}$, i.e. $g$ is the indicator function of the
set~$B$, which satisfies the inequality $\|g-\chi_A\|<\e$ with the
norm of the space $\bar{\Cal H}_G^n$. (Here $\chi_A$ denotes the
indicator function of the set~$A$, and have
$\chi_A\in\bar{\Cal H}_G^n$.) Moreover, if the set $A$ can written
in the form $A=A_1\cup (-A_1)$ with such a set $A_1$ for which the
sets $A_1$ and $-A_1$ have a positive distance from each other,
i.e.\ there is a number $\delta>0$ such that
$\rho(A_1,-A_1)=\inff_{x\in A_1,\,y\in -A_1}\rho(x,y)>\delta$,
where $\rho$ denotes the Euclidean distance in $R^{n\nu}$, then a
good approximation of $\chi_A$ can be given with such a function
$g=\chi_{B\cup(-B)}\in\hat{\bar{\Cal H}}_G^n$ which has some
additional good properties. Namely, there is a set
$B\in\Cal B^{n\nu}$ such that
$B\subset A_1^{\delta/2}=\{x\colon\; \rho(x,A_1)\le\frac\delta2\}$,
$G^n(A_1\,\Delta\,B)<\frac\e2$, where $A\Delta B$ denotes the
symmetric difference of the sets $A$ and $B$, and $G^n$ is the
$n$-times direct product of the spectral measure~$G$ on the space
$R^{n\nu}$, and $g=\chi_{B\cup(-B)}\in\hat{\bar{\Cal H}}_G^n$.
These properties of the set~$B$ imply that the function
$g=\chi_{B\cup(-B)}\in\hat{\bar{\Cal H}}_G^n$ satisfies the
relation $\|g-\chi_A\|<\e$.

To justify this reduction to Statement~A let us observe that if
two functions $f_1\in\bar{\Cal H}_G^n$ and $f_2\in\bar{\Cal H}_G^n$
can be arbitrary well approximated by functions from
$\hat{\bar{\Cal H}}_G^n$ in the $\bar{\Cal H}_G^n$ norm, then the
same relation holds for any linear combination $c_1f_1+c_2f_2$ with
real coefficients~$c_1$ and~$c_2$. (If the functions $f_i$ are
approximated by some functions $g_i\in\hat{\bar{\Cal H}}_G^n$,
$i=1,2$, then we may assume, by applying some refinement of the
partitions if it is necessary, that the regular partitions
appearing in the definition of the approximating functions are
the same.) Hence the proof about the arbitrary good
approximability of a function $f\in\bar{\Cal H}_G^n$ by functions
$g\in\hat{\bar{\Cal H}}_G^n$ can be reduced to the proof about
the arbitrary good approximability of its real part
$\Re f\in\bar{\Cal H}_G^n$ and its imaginary part
$\Im f\in\bar{\Cal H}_G^n$. Moreover, since the real part and
imaginary part of the function~$f$ can be arbitrary well
approximated by such real or imaginary valued functions from
the space $\bar{\Cal H}_G^n$ which take only finitely many values,
the desired approximation result can be reduced to the case when
$f$ is the indicator function of a set $A\in\Cal B^{n\nu}$ such
that $A=-A$ (if $f$ is real valued), or it takes three values,
the value $i$ on a set $A_1\in \Cal B^{n\nu}$, the value~$-i$ on
the set $-A_1$, and it equals zero on
$R^{n\nu}\setminus(A_1\cup(-A_1))$ (if $f$ is purely imaginary valued).
Beside this, the inequalities $G^n(A)<\infty$ and $G^n(A_1)<\infty$
hold. We can even assume that $A$ and $A_1$ are bounded sets,
because $G^n(A)=\limm_{K\to\infty}G^n(A\cap[-K,K]^{n\nu})$, and
the same argument applies for~$A_1$.

Hence Statement~A immediately implies the desired approximation
result in the first case when $f$  is the indicator function of a
set~$A$. In the second case, when such a function~$f$ is
considered that takes the values $\pm i$ and zero, observe that
the sets $A_1=\{x\colon\; f(x)=i\}$ and
$-A_1=\{x\colon\; f(x)=-i\}$ are disjoint. Moreover, we may assume
that they have positive distance from each other, because there
are such compact sets $A_N\subset A$, $N=1,2,\dots$, for which
$\limm_{N\to\infty} G^n(A\setminus A_N)=0$, and two disjoint
compact sets have positive distance. Then the approximation result
also holds in the second case if we take the approximation of the
pair $(A_1,-A_1)$ by the pair $(B,-B)$ appearing in Statement~A,
and define $g(x)=i$ if $x\in B$, $g(x)=-i$ if $x\in-B$ and
$g(x)=0$ otherwise.

In the next step we reduce the proof of Statement~A to the proof of
a result formulated under the name Statement~B. We show that to prove
Property~A it is enough to prove the good approximability of some
very special indicator functions $\chi_B\in\bar{\Cal H}_G^n$ by a
function $g\in\hat{\bar{\Cal H}}_G^n$. We have to handle such sets
$B\in\Cal B^{n\nu}$ where the proof is simpler.

\medskip\noindent
{\it Statement B.}\/ Let $B=D_1\times\cdots\times D_n$ be the direct
product of bounded sets $D_j\in{\Cal B}^\nu$ such that
$D_j\cap(-D_j)=\emptyset$ for all $1\le j\le n$. Then for all
$\e>0$ there is a set $F\subset B\cup(-B)$, $F\in\Cal B^{n\nu}$
such that $\chi_F\in\hat{\bar{\Cal H}}_G^n$, and
$\|\chi_{B\cup(-B)}-\chi_F\|\le\e$, with the norm of the space
$\bar{\Cal H}_G^n$.

\medskip
To deduce Statement~A from statement~B let us first remark
that we may reduce our attention to such sets~$A$ in Statement~A
for which all coordinates of the points of the set~$A$ are
separated from the origin. More explicitly, we may assume the
existence of a number $\eta>0$ with the property
$A\cap K(\eta)=\emptyset$, where $K(\eta)=\bigcupp_{j=1}^n K_j(\eta)$
with $K_j(\eta)=\{(x_1,\dots,x_n)\colon\; x_l\in R^\nu,\;l=1,\dots,n,\;
\rho(x_j,0)\le\eta\}$. To see our right to make such a reduction
observe that the relation $G(\{0\})=0$ implies that
$\limm_{\eta\to0}G^n(K(\eta))=0$, hence
$\limm_{\eta\to0}G^n(A\setminus K(\eta))=G^n(A)$. At this point
we exploited a weakened form of the non-atomic property of the
spectral measure~$G$, namely the relation~$G(\{0\})=0$.

To prove Statement~A with the help of Statement~B it is enough to
show that for all numbers~$\e>0$ and bounded sets
$A\in\Cal B^{n\nu}$ such that $A=-A$ there is a finite sequence
of bounded sets $B_j\in\Cal B^{n\nu}$, $j=\pm1,\dots,\pm N$, such
that the sets $B_j$ are disjoint,
$B_{-j}=-B_j$, $j=\pm1,\dots,\pm N$, each set $B_j$ can be written
in the form $B_j=D^{(j)}_1\times\cdots\times D^{(j)}_n$ with
$D^{(j)}_k\in\Cal B^{\nu}$, and
$D^{(-j)}_k\cap(-D^{(j)}_k)=\emptyset$ for all $1\le j\le N$ and
$1\le k\le n$, and finally the set
$B=\bigcupp_{j=1}^N(B_j\cup B_{-j})$ satisfies the relation
$G^n(A\Delta B)\le\e$. Indeed, since we can choose $\e>0$
arbitrary small, the application of Statement~B for all pairs
$(B_j,-B_j)$ supplies an arbitrary good approximation of the
function $\chi_A$ by a function of the form
$\chi_{\bar B}\in\hat{\bar{\Cal H}}^n_G$
in the norm of the space $\bar{\Cal H}_G^n$.

If the set $A$ can be written in the form $A=A_1\cup(-A_1)$ such
that $\rho(A_1,-A_1)>\delta$, then we can show the existence of a
good approximation of the set~$A$ with the extra properties
formulated in Statement~A in the following way. We may assume that
all sets $B_j$ in the above sequence have a non-empty intersection
with the set~$A$. Otherwise the pair $(B_j,B_{-j})$ could have been
omitted from this sequence. We may also assume, by applying a
refinement of the sets~$B_j$ if it is necessary that all sets
$B_j$ have a diameter less that $\frac\delta4$. Then for a
pair $(B_j,B_{-j})$ one of these sets has a non-empty intersection
with $A_1$ and an empty intersection with $-A_1$, while the
other set has a non-empty intersection with~$-A_1$ and an empty
intersection with~$A_1$. Take the indexes of these sets so that
$B_j\cap A_1\neq\emptyset$. Then it is not difficult to see that
the application of Statement~B for the pairs $(B_j,B_{-j})$ with
such an indexation supplies this part of Statement~A.

To find a sequence~$B_j$ with the above properties for a set~$A$
satisfying the conditions of Statement~A observe that there is a
sequence of finitely many bounded sets $B_j$ of the form
$B_j=D^{(j)}_1\times\cdots\times D^{(j)}_n$,
$D^{(j)}_l\in\Cal B^\nu$, whose union $B=\bigcup B_j$
satisfies the relation $G^{(n)}(A\,\Delta\,B)<\frac\e2$.
Because of the symmetry property $A=-A$ of the set~$A$ we may
assume that these sets $B_j$ have such an indexation with both
positive and negative integers for which $B_j=-B_{-j}$. We may
also demand that $B_j\cap A\neq\emptyset$ for all sets~$B_j$.
Beside this, we may assume, by dividing the sets $D^{(j)}_l$
appearing in the definition of the sets~$B_j$ into smaller sets
if this is needed that their diameter
$\rho(D^{(j)}_l)=\supp_{x\in D_l^{(j)}}\rho(x,0)<\frac\eta2$.
This implies because of the relation $A\cap K(\eta)=\emptyset$
that $D^{(j)}_l\cap(-D^{(j)}_l)=\emptyset$ for all $1\le l\le n$.
Because of these properties of the sets~$B_j$ it can be seen that
with the help of their appropriate further splitting the set~$B$
can be represented as the union of disjoint sets~$B_j$ indexed by
some numbers $j=\pm1,\dots,\pm N$ such that $B_j=-B_{-j}$ for all
$1\le j\le N$, and the pairs $(B_j,B_{-j})$ have the additional
property $D^{(j)}_l\cap(-D^{(j)}_l)=\emptyset$ for all
$1\le l\le n$, and this is what we had to show. For the sake of
completeness we present a partition of the set $B$ with the
properties we need.

Let us first take the following partition of $R^\nu$ for all
$1\le l\le n$. For a fixed number~$l$ this partition consists of
all sets $\bar D^{(l)}_r$ of the form
$\bigcapp_{j>0} F^{r(j)}_{l,j}$, where $r(j)=1,2$ or~3, and
$F^{(1)}_{l,j}=D^{(j)}_l$, $F^{(2)}_{l,j}=-D^{(j)}_l$,
$F^{(3)}_{l,j}=R^\nu\setminus(D^{(j)}_l\cup(-D^{(j)}_l))$.
Then $B$ can be represented as the union of those sets of the
form $\bar D^{(1)}_{r_1}\times\cdots\times\bar D^{(n)}_{r_n}$
which are contained in~$B$.

To prove Statement $B$ first we show that for all $\bar\e>0$ there
is a regular system $\Cal D=\{\Delta_l,\,l=\pm1,\dots,\pm N\}$
such that all sets $D_j$ and $-D_j$, $1\le j\le n$, can be
expressed as the union of some elements $\Delta_l$ of $\Cal D$,
and $G(\Delta_l)\le\bar\e$ for all $\Delta_l\in\Cal D$.

First we show that there is a regular system
$\bar{\Cal D}=\{\Delta'_l,\, l=\pm1,\dots,\pm N'\}$ such that all
sets $D_j$ and $-D_j$ can be expressed as the union of some sets
$\Delta'_l$ of $\bar{\Cal D}$. But we say nothing about the
measure~$G(\Delta'_l)$ of the elements of this regular system. To
get such a regular system we define the sets
$\Delta'(\e_s,\,1\le |s|\le n)
=D^{\e_1}_1\cap(-D_1)^{\e_{-1}}\cap\cdots\cap D^{\e_n}_n\cap(-D_n)^{\e_{-n}}$
for all vectors $(\e_s,\,1\le|s|\le n)$ such that  $\e_s=\pm1$
for all $1\le |s|\le n$, and the vector $(\e_s,\,1\le |s|\le n)$
contains at least  one coordinate~$+1$, and $D^1=D$,
$D^{-1}=R^\nu\setminus D$ for all sets $D\in\Cal B^\nu$.
Then taking an appropriate reindexation of the sets
$\Delta'(\e_s,\,1\le|s|\le n)$ we get a regular system
$\bar{\Cal D}$ with the desired properties. (In this
construction the sets $\Delta'(\e_s,\,1\le|s|\le n)$ are disjoint,
and during their reindexation we drop those of them which equal the
empty set.) To see that $\bar{\Cal D}$ with a good indexation is a
regular system observe that for a set
$\Delta_l=\Delta'(\e_s,\,1\le|s|\le n)\in\bar{\Cal D}$ we have
$-\Delta_l=\Delta'(\e_{-s},\,1\le|s|\le n)\in\bar{\Cal D}$, and
$\Delta_l\cap(-\Delta_l)\subset D_j\cap(-D_j)=\emptyset$ with some
index $1\le j\le n$. (We had to exclude the possibility
$\Delta_l=-\Delta_l$.)

Next we show that by appropriately refining the above regular
system $\bar{\Cal D}$ we can get such a regular system
$\Cal D=\{\Delta_l,\,l=\pm1,\dots,\pm N\}$ which satisfies also
the property $G(\Delta_l)\le\bar\e$ for all $\Delta_l\in\Cal D$.
To show this let us observe that there is a finite partition
$\{E_1,\dots,E_l\}$ of $\bigcupp_{j=1}^n(D_j\cup(-D_j))$ such that
$G(E_j)\le\bar\e$ for all $1\le j \le l$. Indeed, the closure
of $D=\bigcupp_{j=1}^n(D_j\cup(- D_j))$ can be covered by open
sets $H_i\subset R^\nu$ such that $G(H_i)\le\bar\e$ for all sets
$H_i$ because of the non-atomic property of the measure~$G$, and
by the Heyne--Borel theorem this covering can be chosen finite.
With the help of these sets $H_i$ we can get a partition
$\{E_1,\dots,E_l\}$ of $\bigcupp_{j=1}^n(D_j\cup(-D_j))$
with the desired properties.

Then we can make the following construction with the help of the
above sets~$E_j$. Take a pair of elements
$(\Delta'_l,\Delta'_{-l})=(\Delta'_l,-\Delta'_l)$,
of $\bar{\Cal D}$, and split up the
set $\Delta'_l$ with the help of the sets~$E_j$
to the union of finitely many disjoint sets of the form
$\Delta_{l,j}=\Delta'_l\cap E_j$. Then
$G(\Delta_{l,j})<\bar\e$ for all sets $\Delta_{l,j}$, and
we can write the set $\Delta'_{-l}$ as the union of the
disjoint sets $-\Delta_{l,j}$. By applying this procedure for all
pairs $(\Delta'_l,\Delta'_{-l})$ and by reindexing the sets
$\Delta_{l,j}$ obtained by this procedure in an appropriate way we
get a regular system $\Cal D$ with the desired properties.

Let us write $B\cup(-B)$ as the union of products of sets of
the form $\Delta_{l_1}\times\cdots\times\Delta_{l_n}$ with sets
$\Delta_{l_j}\in\Cal D$, $1\le j\le n$, and let us discard those
products for which $l_j=\pm l_{j'}$ for some pair $(j,\,j')$,
$j\neq j'$. We define the set $F$ about which we claim that it
satisfies Property~B as the union of the remaining sets
$\Delta_{l_1}\times\cdots\times\Delta_{l_n}$. Then
$\chi_F\in\hat{\bar{\Cal H}}_G^n$. Hence to prove that
Statement~B holds with this set~$F$ if $\bar\e>0$ is chosen
sufficiently small it is enough to show that the sum of the terms
$G(\Delta_{l_1})\cdots G(\Delta_{l_n})$ for which $l_j=\pm l_{j'}$
with some $j\neq j'$ is less than $n^2\bar\e M^{n-1}$, where
$M=\max G(D_j\cup(-D_j))=2\max G(D_j)$. To see this
observe that for a fixed pair $(j,j')$, $j\neq j'$, the sum
of all products $G(\Delta_{l_1})\cdots G(\Delta_{l_n})$ such that
$l_j=l_{j'}$ can be bounded by $\bar\e M^{n-1}$, and the same
estimate holds if summation is taken for products with the
property $l_j=-l_{j'}$. Indeed, each term of this sum can be
bounded by
$\bar\e G^{n-1}\(\prodd_{1\le p\le n,\,p\neq j}\Delta_{l_p}\)$, and
the events whose $G^{n-1}$ measure is considered in the
investigated sum are disjoint. Beside this their union is in the
product set $\prod\limits_{1\le p\le n,\,p\neq j}(D_p\cup D_{-p})$,
whose measure is bounded by $M^{n-1}$.

\medskip
As the transformation $I_G(f)$ is a contraction from
$\hat{\bar{\Cal H}}_G^n$ into $L_2(G,\Cal A,P)$, it can uniquely
be extended to the closure of $\hat{\bar{\Cal H}}_G^n$, i.e. to
$\bar{\Cal H}_G^n$. We define the $n$-fold Wiener--It\^o integral
in the general case via this extension. The expression $I_G(f)$ is
a real valued random variable for all $f\in\bar{\Cal H}_G^n$, and
relations (4.3), (4.5), $(4.5')$ remain valid for
$f,\,f'\in\bar{\Cal H}_G^n$ or $f\in\Cal H_G^n$ instead of
$f,\,f'\in\hat{\bar{\Cal H}}_G^n$ of $f\in\hat{\Cal H}_G^n$.
Relations $(4.5')$ and (4.6) imply that the transformation
$I_G\colon\; \text{Exp}\,{\Cal H}_G\to L^2(\Omega,\Cal A,P)$ is
an isometry. We shall show that also the following result holds.

\medskip\noindent
{\bf Theorem 4.1.} {\it Let a stationary Gaussian random field be given
(discrete or generalized one), and let $Z_G$ denote the
random spectral measure adapted to it. If we integrate with respect
to this $Z_G$, then the transformation
$I_G\colon\; \text{\rm Exp}\,{\Cal H}_G\to \Cal H$ is unitary. The
transformation $(n!)^{1/2}I_G\colon\; \Cal H_G^n\to\Cal H_n$
is also unitary.}

\medskip
In the proof of Theorem~4.1 we need an identity whose proof is
postponed to the next section.

\medskip\noindent
{\bf Theorem 4.2. (It\^o's formula)} {\it Let $\varphi_1,\dots,\varphi_m$,
\; $\varphi_j\in\Cal H_G^1$, $1\le j\le m$, be an orthonormal system
in $L_G^2$. Let some positive integers $j_1,\dots,j_m$ be given, and
let $j_1+\cdots+j_m=N$. Define for all $i=1,\dots,N$ the function
$g_i$ as $g_i=\varphi_s$ for $j_1+\cdots+j_{s-1}<i\le j_1+\cdots+j_s$,
$1\le s\le m$. (In particular, $g_i=\varphi_1$ for $0<i\le j_1$.) Then
$$ \allowdisplaybreaks
\align
&H_{j_1}\(\int\varphi_1(x)Z_G(\,dx)\)\cdots
H_{j_m}\(\int\varphi_m(x)Z_G(\,dx)\)\\
&\qquad=\int g_1(x_1)\cdots g_N(x_N)\,Z_G(\,dx_1)\cdots Z_G(\,dx_N)\\
&\qquad=\int \Sym[ g_1(x_1)\cdots g_N(x_N)]\,Z_G(\,dx_1)\cdots Z_G(\,dx_N)
\endalign
$$
($H_j(x)$ denotes again the $j$-th Hermite polynomial with leading
coefficient~1.)}

\medskip\noindent
{\it Proof of Theorem 4.1.}\/
The one-fold integral $I_G(f)$, $f\in\Cal H_G^1$, agrees with the
stochastic integral $I(f)$ defined in Section~3. Hence
$I_G(e^{i(n,x)})=X(n)$ in the discrete field case, and
$I_G(\tilde\varphi)=X(\varphi)$, $\varphi\in\Cal S$, in the
generalized field case. Hence $I_G\colon\;\Cal H_G^1\to\Cal H_1$ is
a unitary transformation. Let $\varphi_1,\varphi_2,\dots$ be a
complete orthonormal basis in $\Cal H_G^1$. Then
$\xi_j=\int\varphi_j(x)\,Z_G(\,dx)$, $j=1,2,\dots$, is a complete
orthonormal basis in $\Cal H_G^1$. It\^o's formula implies that
for all sets of positive integers $(j_1,\dots,j_m)$ the random
variable $H_{j_1}(\xi_1)\cdots H_{j_m}(\xi_m)$ can be written as a
$j_1+\cdots+j_m$-fold Wiener--It\^o integral. Therefore
Theorem~2.1 implies that the image of $\text{Exp}\,\Cal H_G$ is the
whole space $\Cal H$, and $I_G\colon\;\text{Exp}\, \Cal H_G$ is
unitary.

The image of $\Cal H_G^n$ contains $\Cal H_n$, because of
Corollary~2.4 and It\'o's formula. Since these images are orthogonal
for different~$n$, formula~(2.1) implies that the image of
$\Cal H_G^n$ coincides with $\Cal H_n$. Hence
$(n!)^{1/2}I_G\colon\; \Cal H_G^n\to\Cal H_n$ is a unitary
transformation.

\medskip
The next result describes the action of shift transformations in
$\Cal H$. We know by Theorem~4.1 that all $\eta\in\Cal H$ can be
written in the form
$$
\eta=f_0+\sum_{n=1}^\infty\frac1{n!}\int f_n(x_1,\dots,x_n)Z_G(\,dx_1)
\dots Z_G(\,dx_n) \tag4.7
$$
with $f=(f_0,f_1,\dots)\in\text{Exp}\,\Cal H_G$ in a unique way, where
$Z_G$ is the random measure adapted to the stationary Gaussian field.

\medskip\noindent
{\bf Theorem 4.3.} {\it Let $\eta\in\Cal H$ have the form~(4.7). Then
$$
T_t\eta=f_0+\sum_{n=1}^\infty\frac1{n!}\int e^{i(t,x_1+\cdots+x_n)}
f_n(x_1,\dots,x_n)Z_G(\,dx_1)
\dots Z_G(\,dx_n)
$$
for all $t\in R^\nu$ in the generalized field and for all
$t\in\text{\BBB Z}_\nu$ in the discrete field case.}

\medskip\noindent
{\it Proof of Theorem 4.3.} Because of formulas~(3.6) and~$(3.6')$ and
the definition of the shift operator~$T_t$ we have
$$
T_t\(\int e^{i(n,x)}Z_G(\,dx)\)=T_tX_n=X_{n+t}
=\int e^{i(t,x)}e^{i(n,x)}Z_G(\,dx), \quad t\in\text{\BBB Z}_\nu,
$$
and because of the identity
$\widetilde{T_t\varphi}(x)=\int e^{(i(u,x)}\varphi(u-t)\,du
=e^{i(t,x)}\tilde\varphi(x)$ for $\varphi\in\Cal S$
$$
T_t\(\int\tilde\varphi(x)\,Z_G(\,dx)\)=T_tX(\varphi)=X(T_t\varphi)
=\int e^{i(t,x)}\tilde\varphi(x)Z_G(\,dx), \quad \varphi\in\Cal S,
\quad t\in R^\nu,
$$
in the discrete and generalized field cases respectively. Hence
$$
T_t\(\int f(x) Z_G(\,dx)\)=\int e^{i(t,x)}f(x) Z_G(\,dx) \quad
\text{if } f\in \Cal H^1_G
$$
for all $t\in\text{\BBB Z}_\nu$ in the discrete field and for all
$t\in R^\nu$ in the generalized field case. This means that
Theorem~4.3 holds in the special case when $\eta$ is a one-fold
Wiener--It\^o integral. Let $f_1(x),\dots,f_m(x)$ be an
orthogonal system in $\Cal H^1_G$. The set of functions
$e^{i(t,x)}f_1(x),\dots,e^{i(t,x)}f_m(x)$ is also an orthogonal
system in $\Cal H^1_G$. ($t\in\text{\BBB Z}_\nu$ in the discrete and
$t\in R^\nu$ in the generalized field case.) Hence It\^o's formula
implies that Theorem~4.3 also holds for random variables of the
form
$$
\eta=H_{j_1}\(\int f_1(x)Z_G(\,dx)\)\cdots
H_{j_m}\(\int f_m(x)Z_G(\,dx)\)
$$
and for their finite linear combinations. Since these linear
combinations are dense in $\Cal H$, Theorem~4.3 holds true.

\medskip
The next result is a formula for the change of variables in
Wiener--It\^o integrals.

\medskip\noindent
{\bf Theorem~4.4.} {\it Let $G$ and $G'$ be two non-atomic spectral
measures such that $G$ is absolutely continuous with respect to $G'$,
and let $g(x)$ be a complex valued function such that
$$
\align
g(x)&=\overline{g(-x)}, \\
|g^2(x)|&=\frac{dG(x)}{dG'(x)}.
\endalign
$$
For every $f=(f_0,f_1,\dots)\in\text{\rm Exp}\,\Cal H_G$, we define
$$
f'_n(x_1,\dots,x_n)=f_n(x_1,\dots,x_n)g(x_1)\cdots g(x_n),
\quad n=1,2,\dots,\quad f'_0=f_0.
$$
Then $f'=(f'_0,f'_1,\dots)\in \text{\rm Exp}\,\Cal H_{G'}^n$, and
$$
\align
&f_0+\sum_{n=1}^\infty\int\frac1{n!}f_n(x_1,\dots,x_n)
Z_G(\,dx_1)\dots Z_G(\,dx_n) \\
&\qquad \overset\Delta\to=f'_0+\sum_{n=1}^\infty\frac1{n!}
\int f_n'(x_1,\dots,x_n)Z_{G'}(\,dx_1) \dots Z_{G'}(\,dx_n),
\endalign
$$
where $Z_G$ and $Z_{G'}$ are Gaussian random spectral measures
corresponding to $G$ and~$G'$.}

\medskip\noindent
{\it Proof of Theorem 4.4.} We have $\|f'_n\|_{G'}=\|f_n\|_G$, hence
$f'\in\text{Exp}\,\Cal H_{G'}$. Let $\varphi_1,\varphi_2,\dots$ be a
complete orthonormal system in $\Cal H_G^1$. Then
$\varphi'_1,\varphi_2',\dots$, \ $\varphi'_j(x)=\varphi_j(x)g(x)$ for
all $j=1,2,\dots$ is a complete orthonormal system in $\Cal H^1_{G'}$.
All functions $f_n\in\Cal H_G^n$ can be written in the form
$f(x_1,\dots,x_n)=\sum c_{j_1,\dots,j_n}\Sym(\varphi_{j_1}(x_1)\cdots
\varphi_{j_n}(x_n))$. Then
$f'(x_1,\dots,x_n)=\sum c_{j_1,\dots,j_n}\Sym(\varphi'_{j_1}(x_1)\cdots
\varphi'_{j_n}(x_n))$. Rewriting all terms
$$
\int\Sym(\varphi_{j_1}(x_1)\cdots\varphi_{j_n}(x_n))
Z_G(\,dx_1)\dots Z_G(,dx_n)
$$
and
$$
\int\Sym(\varphi'_{j_1}(x_1)\cdots\varphi'_{j_n}(x_n))
Z_{G'}(\,dx_1)\dots Z_{G'}(,dx_n)
$$
by means of It\^o's formula we
get that $f$ and $f'$ depend on a sequence of independent standard
normal random variables in the same way. Theorem~4.4 is proved.

\medskip
For the sake of completeness I present in the next Lemma~4.5 another
type of change of variable result. I formulate it only in that simple
case in which we need it in some later considerations.

\medskip\noindent
{\bf Lemma 4.5.} {\it Define for all $t>0$ the (multiplication)
transformation $T_tx=tx$ either from $R^\nu$ to $R^\nu$ or from the
torus $[-\pi,\pi)^\nu$ to the torus $[-\pi t,\pi t)^\nu$. Given a
spectral measure $G$ on $R^\nu$ or on $[-\pi,\pi)^\nu$ define the
spectral measure $G_t$ on $R^\nu$ or on $[-\pi t,\pi t)^\nu$ by the
formula $G_t(A)=G(\frac At)$ for all measurable sets~$A$, and
similarly define the function
$f_{k,t}(x_1,\dots,x_k)=f_k(tx_1,\dots,tx_k)$ for all measurable
functions~$f_k$ of $k$ variables, $k=1,2,\dots$, with
$x_j\in R^\nu$ or $x_j\in [-\pi,\pi)^\nu$ for all $1\le j\le k$,
and put $f_{0,t}=f_0$. If
$f=(f_0,f_1,\dots)\in\text{\rm Exp}\,\Cal H_G$, then
$f_t=(f_{0,t},f_{1,t},\dots)\in\text{\rm Exp}\,\Cal H_{G_t}$, and
$$
\align
&f_0+\sum_{n=1}^\infty\int\frac1{n!}f_n(x_1,\dots,x_n)
Z_G(\,dx_1)\dots Z_G(\,dx_n) \\
&\qquad \overset\Delta\to=f_{0,t}+\sum_{n=1}^\infty\frac1{n!}
\int f_{n,t}(x_1,\dots,x_n)Z_{G_t}(\,dx_1) \dots Z_{G_t}(\,dx_n),
\endalign
$$
where $Z_G$ and $Z_{G_t}$ are Gaussian random spectral measures
corresponding to $G$ and~$G'$.}

\medskip\noindent
{\it Proof of Lemma~4.5.}\/ It is easy to see that
$f_t=(f_{0,t},f_{1,t},\dots)\in\text{\rm Exp}\,\Cal H_{G_t}$. Moreover,
we may define the random spectral measure $Z_{G_t}$ in the
identity we want to prove by the formula $Z_{G_t}(A)=Z_G(\frac At)$.
But with such a choice of $Z_{G_t}$ we can write even $=$ instead of
$\overset\Delta\to=$ in this formula.

\medskip
The next result shows a relation between Wick polynomials and
Wiener--It\^o integrals.

\medskip\noindent
{\bf Theorem 4.6.} {\it Let a stationary Gaussian field be given, and
let $Z_G$ denote the random spectral measure adapted to it. Let
$P(x_1,\dots,x_m)=\sum c_{j_1,\dots,j_n}x_{j_1}\cdots x_{j_n}$
be a homogeneous polynomial of degree~$n$, and let
$h_1,\dots,h_m\in\Cal H_G^1$. (Here $j_1,\dots,j_n$ are $n$ indices
such that $1\le j_l\le m$ for all $1\le l\le n$. It is possible that
$j_l=j_{l'}$ also if $l\neq l'$.) Define the random variables
$\xi_j=\int h_j(x)Z_G(\,dx)$, $j=1,2,\dots,m$, and the function
$\tilde P(u_1,\dots,u_n)=\sum c_{j_1,\dots,j_n} h_{j_1}(u_1)\cdots
h_{j_n}(u_n)$. Then
$$
\:\!P(\xi_1,\dots,\xi_m)\!\:=\int \tilde P(u_1,\dots,u_n)
Z_G(\,du_1)\dots Z_G(\,du_n).
$$
}

\medskip\noindent
{\it Remark.} If $P$ is a polynomial of degree $n$, then it can be
written as $P=P_1+P_2$, where $P_1$ is a homogeneous polynomial of
degree~$n$, and $P_2$ is a polynomial of degree less than~$n$.
Obviously,
$$
\:\!P(\xi_1,\dots,\xi_m)\!\:=\:\!P_1(\xi_1,\dots,\xi_m)\!\:
$$

\medskip\noindent
{\it Proof of Theorem 4.6.}\/ It is enough to show that
$$
\:\!\xi_{j_1}\cdots\xi_{j_n}\!\:=\int h_{j_1}(u_1)\cdots h_{j_n}(u_n)
Z_G(\,du_1)\dots Z_G(\,du_n).
$$
If $h_1,\dots,h_m\in\Cal H_G^1$ are orthonormal, (all functions $h_l$
have norm~1, and if $l\neq l'$, then $h_l$ and $h_{l'}$ are either
orthogonal or $h_l=h_{l'}$), then this relation follows from a comparison
of Corollary~2.3 with It\^o's formula. In the general case an orthonormal
system $\bar h_1,\dots,\bar h_m$ can be found such that
$$
h_j=\sum_{k=1}^m c_{j,k}\bar h_k,\quad j=1,\dots,m
$$
with some real constants $c_{j,k}$. Set $\eta_k=\int \bar h_jZ_G(\,dx)$.
Then
$$
\align
\:\!\xi_{j_1}\cdots\xi_{j_n}\!\:&=\:\! \(\sum_{k=1}^m c_{j_1,k}\eta_k\)
\cdots\(\sum_{k=1}^m c_{j_n,k}\eta_k\)\!\:
=\sum_{k_1,\dots,k_n} c_{j_1,k_1}\cdots c_{j_n,k_n}
\:\!\eta_{k_1}\cdots\eta_{k_n}\!\: \\
&=\sum_{k_1,\dots,k_n} c_{j_1,k_1}\cdots c_{j_n,k_n}
\int \bar h_{k_1}(u_1)\cdots\bar h_{k_n}(u_n)Z_G(\,du_1)\dots Z_G(du_n)\\
&=
\int h_{j_1}(u_1)\cdots h_{j_n}(u_n)Z_G(\,du_1)\dots Z_G(du_n)
\endalign
$$
as we claimed.

\medskip
We finish this section by showing how the Wiener--It\^o integral can
be defined if the spectral measure~$G$ may have atoms. We do this
although such a construction seems to have a limited importance as
in most applications the restriction that we apply the Wiener--It\^o
integral only in the case of a non-atomic spectral measure~$G$
causes no serious problem. If we try to give this definition by
modifying the original one, then we have to split up the atoms. The
simplest way we found for this splitting up, was the use
of randomization.

Let $G$ be a spectral measure on $R^\nu$, and let $Z_G$ be  a
corresponding  Gaussian spectral random measure on a probability
space $(\Omega,\Cal A,P)$. Let us define a new spectral measure
$\hat G=G\times\lambda_{[-\frac12,\frac12]}$ on $R^{\nu+1}$, where
$\lambda_{[-\frac12,\frac12]}$ denotes the uniform distribution on the
interval $[-\frac12,\frac12]$. If the probability space
$(\Omega,\Cal A,P)$ is sufficiently rich, a random spectral measure
$Z_{\hat G}$ corresponding to $\hat G$ can be defined on it in such a
way that $Z_{\hat G}(A\times [-\frac12,\frac12])=Z_G(A)$ for
all $A\in\Cal B^\nu$. For $f\in \bar{\Cal H}_G^n$ we define the function
$\hat f\in\bar{\Cal H}_{\hat G}^n$ by the formula
$\hat f(y_1,\dots,y_n)=f(x_1,\dots,x_n)$ if $y_j$ is the juxtaposition
$(x_j,u_j)$, \ $x_j\in R^\nu$, $u_j\in R^1$, $j=1,2,\dots,n$. Finally
we define the Wiener--It\^o integral in the general case by the
formula
$$
\int f(x_1,\dots,x_n)Z_G(\,dx_1)\dots Z_G(\,dx_n)
=\int \hat f(y_1,\dots,y_n)Z_{\hat G}(\,dy_1)\dots Z_{\hat G}(dy_n).
$$
(What we actually have done was to introduce a virtual new
coordinate~$u$. With the help of this new coordinate we could reduce
the general case to the special case when $G$ is non-atomic.) If $G$
is a non-atomic  spectral measure, then the new definition of
Wiener--It\^o integrals coincides with the original one. It is easy to
check this fact for one-fold integrals, and then It\^o's formula
proves it for multiple integrals. It can be seen with the help of
It\^o's formula again, that all results of this section remain valid
for the new definition of Wiener--It\^o integrals. In particular, we
formulate the following result.

Given a stationary Gaussian field let $Z_G$ be the random spectral
measure adapted to it. All $f\in\Cal H_G^n$ can be written in the
form
$$
f(x_1,\dots,x_n)=\sum
c_{j_1,\dots,j_n}\varphi_{j_1}(x_1)\cdots\varphi_{j_n}(x_n) \tag4.8
$$
with some functions $\varphi_j\in\Cal H_G^1$, $j=1,2,\dots$. Define
$\xi_j=\int\varphi_j(x)Z_G(\,dx)$. If $f$ has the form~(4.8), then
$$
\int f(x_1,\dots,x_n)Z_G(\,dx_1)\dots Z_G(\,dx_n)=
\sum c_{j_1,\dots,j_n}\:\!\xi_{j_1}\cdots\xi_{j_n}\!\:.
$$
The last identity would provide another possibility for defining
Wiener--It\^o integrals.

\vfill\eject

\item{\bf 5.} {\bf The proof of It\^o's formula. The diagram formula
and some of its consequences.}

\medskip\noindent
We shall prove It\^o's formula with the help of the following

\medskip\noindent
{\bf Proposition 5.1.} {\it Let $f\in\bar{\Cal H}_G^n$ and
$h\in\bar{\Cal H}_G^1$. Let us define the functions
$$
f\underset k\to\times h(x_1,\dots,x_{k-1},x_{k+1},\dots,x_n)
=\int f(x_1,\dots,x_n)\overline{h(x_k)} G(\,dx_k), \quad k=1,\dots,n,
$$
and
$$
fh(x_1,\dots,x_{n+1})=f(x_1,\dots,x_n)h(x_{n+1}).
$$
Then $f\underset k\to\times h$, $k=1,\dots,n$, and $fh$ are in
$\bar{\Cal H}_G^{n-1}$ and $\bar{\Cal H}_G^{n+1}$ respectively, and
their norm satisfies the inequality
$\|f\underset k\to\times h\|\le\|f\|\cdot\|h\|$ and
$\|fh\|\le\|f\|\cdot\|h\|$. The relation
$$
n!I_G(f)I_G(h)=(n+1)!I_G(fh)+\sum_{k=1}^n (n-1)!
I_G\(f\underset k\to\times h\)
$$
holds true.}

\medskip\noindent
{\it Remark.}\/ There is a small inaccuracy in the formulation of
Lemma~5.1. We considered the Wiener--It\^o integral of the function
$f\underset k\to\times h$ with arguments
$x_1,\dots,x_{k-1},x_{k+1},\dots,x_n$, while we defined this integral
for functions with arguments $x_1,\dots,x_{n-1}$. We can correct this
inaccuracy by reindexing the variables of
$f\underset k\to\times h$ and to work with the function
$(f\underset k\to\times h)'(x_1,\dots,x_{n-1})=
f\underset k\to\times h(x_{\alpha_k(1)},\dots,x_{\alpha_k(k-1)},
x_{\alpha_k(k+1)},\dots,x_{\alpha_k(n)})$ instead of
$f\underset k\to\times h$, where  $\alpha_k(j)=j$ for
$1\le j\le k-1$ and $\alpha_k(j)=j-1$ for $k+1\le j\le n$.

\medskip
We also need the following recursion formula for Hermite polynomials.

\medskip\noindent
{\bf Lemma 5.2.} {\it
$$
H_n(x)=xH_{n-1}(x)-(n-1)H_{n-2}(x) \quad\text{for \ }n=1,2,\dots,
$$
with the notation $H_{-1}(x)\equiv0$.}

\medskip\noindent
{\it Proof of Lemma 5.2.}
$$ \allowdisplaybreaks
\align
H_n(x)=(-1)^ne^{x^2/2}\frac{d^n}{dx^n}(e^{-x^2/2})
&=-e^{x^2/2}\frac d{dx}\(H_{n-1}(x)e^{-x^2/2}\)\\
&=xH_{n-1}(x)-\frac d{dx}H_{n-1}(x).
\endalign
$$
Since $\frac d{dx}H_{n-1}(x)$ is a polynomial of order $n-2$ with leading
coefficient $n-1$ we can write
$$
\frac d{dx} H_{n-1}(x)=(n-1)H_{n-2}(x)+\sum_{j=0}^{n-3}c_jH_j(x).
$$
To complete the proof of Lemma~5.2 it remains to show that in the last
expansion all coefficients~$c_j$ are zero. This follows from the
orthogonality of the Hermite polynomials and the calculation
$$
\align
\int e^{-x^2/2}H_j(x)\frac d{dx}H_{n-1}(x)\,dx
&=-\int H_{n-1}(x)\frac d{dx}(e^{-x^2/2}H_j(x))\,dx\\
&=\int e^{-x^2/2}H_{n-1}(x)P_{j+1}(x)\,dx=0
\endalign
$$
with the polynomial $P_{j+1}(x)=xH_j(x)-\frac d{dx}H_j(x)$ of
order~$j+1$ for $j\le n-3$.

\medskip\noindent
{\it Proof of Theorem 4.2 via Proposition 5.1.}\/ We prove Theorem~4.2
by induction. Theorem~4.2 holds for $N=1$. Assume that it holds
for~$N-1$. Let us define the functions
$$
\align
f(x_1,\dots,x_{N-1})&=g_1(x_1)\cdots g_{N-1}(x_{N-1})\\
h(x)&=g_N(x).
\endalign
$$
Then
$$
\align
J&=\int g_1(x_1)\cdots g_N(x_N)Z_G(\,dx_1)\dots Z_G(\,dx_N)\\
&=N!\,I_G(fh)=(N-1)!\,I_G(f)I_G(h)-\sum_{k=1}^{N-1} (N-2)!\,
I_G\(f\underset k\to\times h\)
\endalign
$$
by Proposition~5.1. The induction hypothesis implies that
$$
\align
J=&H_{j_1}\(\int\varphi_1(x)Z_G(\,dx)\)\cdots
H_{j_{m-1}}\(\int\varphi_{m-1}(x)Z_G(\,dx)\) \\
&\qquad\qquad\qquad H_{j_m-1}\(\int\varphi_{m}(x)Z_G(\,dx)\)
\int\varphi_m(x)Z_G(\,dx)\\
&\qquad-(j_m-1)H_{j_1}\(\int\varphi_1(x)Z_G(\,dx)\)\cdots
H_{j_{m-1}}\(\int\varphi_{m-1}(x)Z_G(\,dx)\)\\
&\qquad\qquad\qquad H_{j_m-2}\(\int\varphi_m(x)Z_G(\,dx)\),
\endalign
$$
where $H_{j_m-2}(x)=H_{-1}(x)\equiv0$ if $j_m=1$. This relation
holds, since
$$
\align
&f\underset k\to\times h(x_1,\dots,x_{k-1},x_{k+1},\dots,x_{N-1})
=\int g_1(x_1)\cdots g_{N-1}(x_{N-1})\overline{\varphi_m(x_k)}G(\,dx_k)\\
&\qquad =
\left\{   \aligned
&0 \quad \text{if \ } k\le N-j_m\\
&g_1(x_1)\cdots g_{k-1}(x_{k-1})g_{k+1}(x_{k+1})\cdots g_{N-1}(x_{N-1})
\quad \text{if } N-j_m<k\le N-1.
\endaligned \right.
\endalign
$$
Hence Lemma~5.2 implies that
$$
\align
J&=\prod_{s=1}^{m-1}H_{j_s}\(\int \varphi_s(x)Z_G(\,dx)\)
\biggl[ H_{j_m-1}\(\int\varphi_m(x)Z_G(\,dx)\)\int\varphi_m(x)Z_G(\,dx) \\
&\qquad -(j_m-1)H_{j_m-2}\(\int\varphi_m(x)Z_G(\,dx)\)\biggr]
=\prod_{s=1}^m H_{j_s}\(\int\varphi_s(x)Z_G(\,dx)\),
\endalign
$$
as claimed.

\medskip
Let us fix some functions $h_1\in\bar{\Cal H}_G^{n_1}$,\dots,
$h_m\in\bar{\Cal H}_G^{n_m}$. In the next result, in the so-called
diagram formula, we express the product
$n_1!I_G(h_1)\cdots n_m!I_G(h_m)$ as the sum of Wiener--It\^o
integrals. This result contains Proposition~5.1 as a special case.
There is no unique terminology for this result in the literature.
We shall follow the notation of Dobrushin in~[7].

We shall use the term diagram of order $(n_1,\dots,n_m)$ for an
undirected graph of $n_1+\cdots+n_m$ vertices such that its vertices
are indexed by the pairs of integers~$(j,l)$, $l=1,\dots,m$, \
$j=1,\dots,n_l$, with the properties that no more than one edge
enters into each vertex, and  edges can connect only pairs of
vertices~$(j_1,l_1)$ and $(j_2,l_2)$ for which $l_1\neq l_2$.
Let $\Gamma=\Gamma(n_1,\dots,n_m)$ denote the set of all diagrams.
Given a diagram $\gamma\in\Gamma$ \ $|\gamma|$ denotes the
number of edges in~$\gamma$. Let there be given a set of functions
$h_1\in\bar{\Cal H}_G^{n_1}$,\dots, $h_m\in\bar{\Cal H}_G^{n_m}$.
Sometimes we denote the variables of the function $h_l$ by $x_{(j,l)}$
instead of $x_j$, i.e. we write $h_l(x_{(1,l)},\dots,x_{(n_l,l)})$
instead of $h_l(x_1,\dots,x_{n_l})$.  Put $N=n_1+\cdots+n_m$. We
introduce the function of $N$~variables corresponding to the
vertices of the diagram by the formula
$$
h(x_{(j,l)},\; l=1,\dots,m,\;j=1,\dots,n_l)=\prod_{l=1}^m
h_l(x_{(j,l)},\;j=1,\dots,n_l).
$$
Fixing a diagram $\gamma\in\Gamma$ we enumerate the variables $x_{(j,l)}$
in such a way that the vertices into which no edges enter will have
the numbers $1,2,\dots,N-2|\gamma|$ and the vertices connected by an
edge will have the numbers~$p$ and $p+|\gamma|$, where
$p=N-2|\gamma|+1,\dots,N-|\gamma|$. Let
$$
\aligned
h_\gamma(x_1,\dots,x_{N-2|\gamma|})&=\idotsint h(x_1,\dots,x_{N-|\gamma|},
-x_{N-2|\gamma|+1},\dots,-x_{N-|\gamma|})\\
&\qquad G(\,dx_{N-2|\gamma|+1})\dots G(\,dx_{N-|\gamma|}).
\endaligned \tag5.1
$$
The function $h_\gamma$ depends only on the variables
$x_1,\dots,x_{N-2|\gamma|}$, i.e. it is
independent of how the vertices connected by edges are indexed. Indeed,
it follows from the evenness of the spectral measure that by
interchanging the indices $s$ and $s+\gamma$ of two vertices connected
by an edge does not change the value of the  integral $h_\gamma$. Let
us now consider $I_G(h_\gamma)$. The function $h_\gamma$ may depend on
the numbering of those vertices of~$\gamma$ from which no edge starts,
but $\Sym h_\gamma$ and therefore $I_G(h_\gamma)$ does not depend on
it. Now we formulate the following

\medskip\noindent
{\bf Theorem 5.3. (Diagram formula)} {\it For all functions
$h_1\in\bar{\Cal H}_G^{n_1}$,\dots, $h_m\in\bar{\Cal H}_G^{n_m}$, \
$n_1,\dots,n_m=1,2,\dots$, the following relations hold:

\medskip
\item{A)} $h_\gamma\in\bar{\Cal H}_G^{n-2|\gamma|}$, and
$\|h_\gamma\|\le\prodd_{j=1}^m\|h_j\|$ for all $\gamma\in\Gamma$.

\item{B)} $n_1!I_G(h_1)\cdots n_m! I_G(h_m)
=\summ_{\gamma\in\Gamma}(N-2|\gamma|)!I_G(h_\gamma)$.}

\medskip\noindent
{\it Remark.}\/ In the special case $m=2$, $n_1=n$, $n_2=1$ Theorem~5.3
coincides with Proposition~5.1. To see this it is enough to observe that
$h(-x)=\overline{h(x)}$ for all $h\in \bar{\Cal H}_G^1$.

\medskip\noindent
{\it Proof of Theorem~5.3.}\/ It suffices to prove Theorem~5.3 in the
special case $m=2$. Then the case $m>2$ follows by induction.

We shall use the notation $n_1=n$, $n_2=m$, and we write
$x_1,\dots,x_{n+m}$ instead of
$x_{(1,1)},\dots,x_{(n,1)},x_{(1,2)}\dots,x_{(m,2)}$. It is clear
that the function $h_\gamma$ satisfies property~(a) of the classes
$\bar{\Cal H}_G^j$ defined in Section~4. We show that Part~A of
Theorem~5.3 is a consequence of the Schwartz inequality.
To prove this estimate on the norm of $h_\gamma$ it is enough
to restrict ourselves to such diagrams~$\gamma$ in which the vertices
$(n,1)$ and $(m,2)$, \ $(n-1,1)$ and $(m-1,2)$,\dots, $(n-k,1)$ and
$(m-k,2)$ are connected by edges with some $0\le k\le \min(n,m)$.
In this case we can write
$$
\align
&|h_\gamma(x_1,\dots,x_{n-k-1},x_{n+1},\dots,x_{n+m-k-1})|^2 \\
&\qquad=\biggl|\int h_1(x_1,\dots,x_n)h_2(x_{n+1},\dots,x_{n+m-k-1},-x_{n-k},
\dots,-x_n)\\
&\qquad\qquad\qquad G(\,dx_{n-k})\dots G(\,dx_n)\biggr|^2 \\
&\qquad\le \int |h_1(x_1,\dots,x_n)|^2 G(\,dx_{n-k})\dots G(\,dx_n) \\
&\qquad\qquad\qquad
\int |h_2(x_{n+1},\dots,x_{n+m})|^2 G(\,dx_{n+m-k})\dots G(\,dx_{n+m})
\endalign
$$
by the Schwartz inequality and the symmetry $G(-A)=G(A)$ of the
spectral measure~$G$. Integrating this inequality with respect to
the free variables we get part~A) of Theorem~5.3.

In the proof of part~B) first we restrict ourselves to the case when
$h_1\in\hat{\bar{\Cal H}}_G^n$ and $h_2\in\hat{\bar{\Cal H}}_G^m$.
Assume that they are adapted to a regular system
$\Cal D=\{\Delta_j,\;j=\pm1,\dots,\pm N\}$ of subsets of $R^n$ with
finite measure~$G$. We may even assume that all $\Delta_j\in\Cal D$
satisfy the inequality $G(\Delta_j)<\e$ with some $\e>0$ to be chosen
later, because otherwise we could split up the sets $\Delta_j$ into
smaller ones. Let us fix a point $u_j\in\Delta_j$ in all sets
$\Delta_j\in\Cal D$. Put $K_i=\supp_x|h_i(x)|$, $i=1,2$, and let $A$
be a cube containing all $\Delta_j$.

We can write
$$
\align
I=n!I_G(h_1)m!I_G(h_2)&={\sum}' h_1(u_{j_1},\dots,u_{j_n})
h_2(u_{k_1},\dots,u_{k_m})\\
&\qquad\qquad Z_G(\Delta_{j_1})\cdots Z_G(\Delta_{j_n})
Z_G(\Delta_{k_1})\cdots Z_G(\Delta_{k_m})
\endalign
$$
with the numbers $u_{j_p}\in\Delta_{j_p}$ and
$u_{k_r}\in\Delta_{k_r}$ we have fixed, where the summation in
$\sum'$ goes through all pairs $((j_1,\dots,j_n),(k_1,\dots,k_m))$,
$j_p,\,k_r\in\{\pm1,\dots,\pm N\}$, \ $p=1,\dots,n$, $r=1,\dots,m$,
such that $j_p\neq\pm j_{\bar p}$ and $k_r\neq\pm k_{\bar r}$ if
$p\neq\bar p$ or $r\neq\bar r$.

Write
$$
\align
I&=\sum_{\gamma\in\Gamma}{\sum}^\gamma \,
h_1(u_{j_1},\dots,u_{j_n}) h_2(u_{k_1},\dots,u_{k_m}) \\
&\qquad\qquad\qquad Z_G(\Delta_{j_1})\cdots Z_G(\Delta_{j_n})
Z_G(\Delta_{k_1})\cdots Z_G(\Delta_{k_m}),
\endalign
$$
where $\sum^\gamma$ contains those terms of $\sum'$ for which
$j_p=k_r$ or $j_p=-k_r$ if the vertices $(1,p)$ and $(2,r)$ are
connected in $\gamma$, and $j_p\neq \pm k_r$ if $(1,p)$ and $(2,r)$
are not connected. Let us define the sets
$$
\align
A_1&=A_1(\gamma)=\{p\colon\;p\in\{1,\dots,n\},\text{ and no
edge starts from }(p,1)\text{ in }\gamma\},\\
A_2&=A_2(\gamma)=\{r\colon\;r\in\{1,\dots,m\},\text{ and no
edge starts from }(r,2)\text{ in }\gamma\}
\endalign
$$
and
$$
\align
B=B(\gamma)&=\{(p,r)\colon\; p\in\{1,\dots,n\},\; r\in\{1,\dots,m\},\\
&\qquad\qquad (p,1) \text{ and } (r,2) \text{ are connected in }\gamma\}
\endalign
$$
together with the map $\alpha\colon\;\{1,\dots,n\}\setminus A_1\to
\{1,\dots,m\}\setminus A_2$ defined as
$$
\alpha(p)=r \quad \text{if } (p,r)\in B \quad \text{for all \ }
p\in\{1,\dots,n\}\setminus A_1. \tag5.2
$$

Let $\Sigma^\gamma$ denote the value of the inner sum $\sum^\gamma$
for some $\gamma\in\Gamma$ in the last summation formula, and
write it in the form
$$
\Sigma^\gamma=\Sigma_1^\gamma+\Sigma_2^\gamma
$$
with
$$
\align
\Sigma_1^\gamma&={\sum}^\gamma h_1(u_{j_1},\dots,u_{j_n})
h_2(u_{k_1},\dots,u_{k_m})
\prod_{p\in A_1}Z_G(\Delta_{j_p})\prod_{r\in A_2} Z_G(\Delta_{k_r})\\
&\qquad\qquad\qquad\cdot
\prod_{(p,r)\in B} E\(Z_G(\Delta_{j_p})Z_G(\Delta_{k_r})\)
\endalign
$$
and
$$
\align
\Sigma_2^\gamma&={\sum}^\gamma h_1(u_{j_1},\dots,u_{j_n})
h_2(u_{k_1},\dots,u_{k_m})
\prod_{p\in A_1}Z_G(\Delta_{j_p})\prod_{r\in A_2} Z_G(\Delta_{k_r})\\
&\qquad \cdot\[\prod_{(p,r)\in B}Z_G(\Delta_{j_p})Z_G(\Delta_{k_r})
-E\( \prod_{(p,r)\in B} Z_G(\Delta_{j_p})Z_G(\Delta_{k_r})\)\].
\endalign
$$
The random variables $\Sigma^\gamma_1$ and $\Sigma^\gamma_2$ are
real valued. To see this observe that if the sum defining these
expressions contains a term with arguments $\Delta_{j_p}$, and
$\Delta_{k_r}$, then it also contains the term with arguments
$-\Delta_{j_p}$ and $-\Delta_{k_r}$. This fact together with
property~(v) of the random spectral measure~$Z_G$ and the
analogous property of the functions~$h_1$  and~$h_2$ imply that
$\Sigma^\gamma_1=\overline{\Sigma^\gamma_1}$ and
$\Sigma^\gamma_2=\overline{\Sigma^\gamma_2}$. Hence these random
variables are real valued. As a consequence, we can bound
$(n+m-2|\gamma|)!I_G(h_\gamma)-\Sigma_1^\gamma$ and
$\Sigma_2^\gamma$ by means of their second moment.

We are going to show that $\Sigma_1^\gamma$ is a good
approximation of $(n+m-2|\gamma|)!\,I_G(h_\gamma)$, and
$\Sigma_2^\gamma$ is negligibly small. This implies that
$(n+m-2|\gamma|)!I_G(h_\gamma)$ well approximates~$\Sigma^\gamma$.

To estimate $(n+m-2|\gamma|)!I_G(h_\gamma)-\Sigma_1^\gamma$ we rewrite
$\Sigma^\gamma_1$ as a Wiener--It\^o integral with a kernel function
adapted to the regular system~$\Cal D$ which is close to~$h_\gamma$.
To find this kernel function we rewrite the sum
defining~$\Sigma^\gamma_1$ by first fixing the variables $u_{j_p}$,
$p\in A_1$, and $u_{k_r}$, $r\in A_2$, and summing up by the
remaining variables, and after this summing by the variables fixed
at the first step. We get that
$$
\aligned
\Sigma_1^\gamma&=\sum\Sb j_p\colon\; 1\le |j_p|\le N
\text{ for all }p\in A_1\\
k_r\colon\; 1\le |k_r|\le N \text{ for all }r\in A_2\endSb
h_{\gamma,1}(j_p,\;p\in A_1,\, k_r,\, r\in A_2)
\prod_{p\in A_1}Z_G(\Delta_{j_p})\prod_{r\in A_2} Z_G(\Delta_{k_r})
\endaligned \tag5.3
$$
with a function $h_{\gamma,1}$ depending on the arguments
$j_p$, $p\in A_1$, and $k_r$, $r\in A_2$, with values
$j_p,k_r\in\{\pm1,\dots,\pm N\}$   defined with the help another
function $h_{\gamma,2}$ described below. It also depends on the
arguments $j_p$, $p\in A_1$, and $k_r$, $r\in A_2$, with values
$j_p,k_r\in\{\pm1,\dots,\pm N\}$. Formula~(5.3) holds with
$$
h_{\gamma,1}(j_p,\,p\in A_1,\;k_r,\,r\in A_2)=0  \tag5.4a
$$
if the numbers of the set
$\{\pm j_p\colon\;p\in A_1\}\cup\{\pm k_r\colon\;r\in A_2\}$
are not all different, and
$$
h_{\gamma,1}(j_p,\,p\in A_1,\;k_r,\,r\in A_2)
=h_{\gamma,2}(j_p,\,p\in A_1,\;k_r,\,r\in A_2) \tag5.4b
$$
if all numbers $\pm j_p$, $p\in A_1$, and $\pm k_r$, $r\in A_2$ are
different with the function
$h_{\gamma,2}(j_p,\,p\in A_1,\;k_r,\,r\in A_2)$ defined for all
sequences $j_p$, $p\in A_1$ and $k_r$, $r\in A_2$, with
$j_p,k_r\in\{\pm1,\dots,\pm N\}$ (i.e. also in the case when some of
the arguments~$j_p$, $p\in A_1$, or $k_r$, $r\in A_2$, agree) by the
formula
$$
\aligned
h_{\gamma,2}(j_p,\;p\in A_1,\,k_r,\,r\in A_2)
&={\sum}^{\gamma,1} h_1(u_{j_1},\dots,u_{j_n})h_2(u_{k_1},\dots,u_{k_m})\\
&\qquad\cdot \prod_{(p,r)\in B}
E\(Z_G(\Delta_{j_p})Z_G(\Delta_{k_r})\).
\endaligned \tag5.5
$$
The summation $\sum^{\gamma,1}$ in formula~(5.5) which depends on
the arguments $j_p$, $p\in A_1$, and $k_r$, $r\in A_2$, is defined
in the following way. We sum up for such sequences $j_p$, $k_r$ with
indices $p\in \{1,\dots,n\}\setminus A_1$ and
$r\in \{1,\dots,m\}\setminus A_2$ which satisfy the following
conditions. Put $C=\{\pm j_p,\,p\in A_1\}\cup\{\pm k_r,\,r\in A_2\}$.
We demand that all numbers $j_p$ and $k_r$ with indices
$p\in\{1,\dots,n\}\setminus A_1$ and $r\in\{1,\dots,m\}\setminus A_2$
are such that $j_p,k_r\in\{\pm1,\dots,\pm N\}\setminus C$. Let us write
all numbers $r\in\{1,\dots,m\}\setminus A_2$ in the form $r=\alpha(p)$,
$p\in\{1,\dots,n\}\setminus A_1$, with the map~$\alpha$ defined
in~(5.2). We also demand that only such $k_r=k_{\alpha(p)}$ appear
in the summation for which $k_{\alpha(p)}=\pm j_p$ for all
$p\in\{1,\dots,n\}\setminus A_1$. Beside this, all numbers $\pm j_p$,
$p\in\{1,\dots,n\}\setminus A_1$, must be different. The summation
in $\sum^{\gamma,1}$ is taken for all such sequences $j_p$,
$p\in\{1,\dots,n\}\setminus A_1$ and $k_r$,
$r\in\{1,\dots,m\}\setminus A_2$ which satisfy the above conditions.

Formula~(5.5) can be rewritten in a simpler form. To do this let
us first observe that the condition $k_{\alpha(p)}=\pm j_p$ can be
replaced by the condition $k_{\alpha(p)}=-j_p$ in it, and we can
write $G(\Delta_{j_p})$ instead of the term
$EZ_G(\Delta_{j_p})Z_G(\Delta_{k_r})$ (with $(p,r)\in B$) in the
product at the end of~(5.5). This follows from the fact that
$EZ_G(\Delta_{j_p})Z_G(\Delta_{k_r})=EZ_G(\Delta_{j_p})^2=0$
if $k_r=j_p$ and $EZ_G(\Delta_{j_p})Z_G(\Delta_{k_r})
=EZ_G(\Delta_{j_p}Z_G(-\Delta_{j_p})=G(\Delta_{j_p})$ if $k_r=-j_p$.
Beside this, the expression in~(5.5) does not change if we take
summation for all sequences $j_p$, $p\in\{1,\dots,n\}\setminus A$,
with $j_p\in\{\pm1,\dots,\pm N\}$, because in such a way we only
attach such terms to the sum which equal zero. This follows from the
fact that both functions $h_1$ and $h_2$ are adapted to the regular
system~$\Cal D$, hence
$h_1(u_{j_1},\dots,u_{j_n})h_2(u_{k_1},\dots,u_{k_m})=0$ if for an
index $p\in\{1,\dots,n\}\setminus A_1$ $j_p=\pm j_{p'}$ with
$p\neq p'$ or $j_p=-k_r$ with $(p,r)\in B$, and beside this there
exists some $r'\in A_2$ such that $j_p=\pm k_{r'}$.

The above relations enable us to rewrite~(5.5) in the following
way. Let us define that map $\alpha^{-1}$ on the set
$\{1,\dots,m\}\setminus A_2$  which is the inverse of the map
$\alpha$ defined in~(5.2), i.e.\ $\alpha^{-1}(r)=p$ if $(p,r)\in B$.
With this notation we can write
$$
\align
&h_{\gamma,2}(j_p,\,p\in A_1,\;k_r,\,r\in A_2)\\
&\qquad= \!\!\!\!\!\!\!
\sum \Sb j_p,\,p\in \{1,\dots,n\}\setminus A_1,\\
1\le |j_p|\le N \text{ for all indices } p \endSb \!\!\!\!\!\!\!\!\!\!
h_1(u_{j_1},\dots,u_{j_n})
h_2(u_{k_r},\,r\in A_2, -u_{j_{\alpha^{-1}(r)}},
\,r\in\{1,\dots,m\}\setminus A_2) \\
&\qquad\qquad\qquad\qquad
\prod_{p\in \{1,\dots,n\}\setminus A_1} G(\Delta_{j_p}). \tag5.6
\endalign
$$
Formula~(5.6) can be rewritten as
$$
\align
&h_{\gamma,2}(j_p,\,p\in A_1,\;k_r,\,r\in A_2)\\
&\qquad=\int h_1(u_{j_p},\,p\in A_1,\;
x_p,\,p\in\{1,\dots,n\}\setminus A_1)\\
&\qquad\qquad h_2(u_{k_r},\,r\in A_2,\; -x_{\alpha^{-1}(r)},
\,r\in\{1,\dots,m\}\setminus A_2)
\prod_{p\in \{1,\dots,n\}\setminus A_1} G(\,dx_p). \tag5.7
\endalign
$$

We define with the help of $h_{\gamma,1}$ and $h_{\gamma,2}$ two
new functions on $R^{(n+m-2|\gamma|)\nu}$ with arguments
$x_1,\dots,x_{n+m-2|\gamma}$. The first one will be the kernel
function of the Wiener--It\^o integral expressing~$\Sigma^\gamma_1$
and the second one will be equal to the function~$h_\gamma$ defined
in~(5.1). We define these functions in two steps. In the first step
we reindex the arguments of the functions~$h_{1,\gamma}$
and~$h_2,\gamma$ to get functions depending on sequences
$j_1,\dots,j_{n+m-2|\gamma|}$. For this goal we list the elements of
the sets $A_1$ and $A_2$ as $A_1=\{p_1,\dots,p_{n-|\gamma|}\}$ with
$1\le p_1<p_2<\cdots<p_{n-|\gamma|}\le n$ and
$A_2=\{r_1,\dots,r_{m-|\gamma|}\}$ with
$1\le r_1<r_2<\cdots<p_{m-|\gamma|}\le m$
and define the  maps $\beta_1\colon\; A_1\to\{1,\dots,n-|\gamma|\}$
and $\beta_2\colon\; A_2\to\{n-|\gamma|+1,\dots,n+m-2|\gamma|\}$
by the formulas $\beta_1(p_l)=l$ if $1\le l\le n-\gamma$,
$1\le l\le n-|\gamma|$, and $\beta_2(r_l)=l+n-|\gamma|$,
$1\le l\le m-|\gamma|$, if $n-|\gamma|+1\le l\le n+m-2|\gamma|$.
We define with the help of the maps~$\beta_1$ and~$\beta_2$ the
functions
$$
h_{\gamma,3}(j_1,\dots,j_{n+m-2|\gamma|})=
h_{\gamma,1}(j_{\beta_1(r_1)},\dots,j_{\beta_1(n-|\gamma|))},
k_{\beta_2(1)},\dots,k_{\beta_2(m-|\gamma|)})
$$
and
$$
h_{\gamma,4}(j_1,\dots,j_{n+m-2|\gamma|})=
h_{\gamma,2}(j_{\beta_1(r_1)},\dots,j_{\beta_1(n-|\gamma|))},
k_{\beta_2(1)},\dots,k_{\beta_2(m-|\gamma|)}),
$$
where the arguments of the functions $h_{\gamma,3}$ and
$h_{\gamma,4}$ are sequences $j_1,\dots,j_{n+m-2|\gamma|}$ with
$j_s\in\{\pm1,\dots,\pm N\}$ for all $1\le s\le n+m-2|\gamma|$.

With the help of the above functions we define the following
functions $h_{\gamma,5}$ and $h_{\gamma,6}$ on
$R^{(n+m-2|\gamma|)\nu}$.
$$
h_{\gamma,5}(x_1,\dots,x_{n+m-2|\gamma|})=\left\{
\aligned
&h_{\gamma,3}(j_1,\dots,j_{n+m-2|\gamma|})
\quad\text{ if } x_l\in \Delta_{j_l}, \\
&\qquad \text{ for all } 1\le l\le n+m-2|\gamma|\\
&0 \quad\text{otherwise,}
\endaligned \right.
$$
and
$$
h_{\gamma,6}(x_1,\dots,x_{n+m-2|\gamma|})=\left\{
\aligned
&h_{\gamma,4}(j_1,\dots,j_{n+m-2|\gamma|})
\quad\text{ if } x_l\in \Delta_{j_l}, \\
&\qquad \text{ for all } 1\le l\le n+m-2|\gamma|\\
&0 \quad\text{otherwise.}
\endaligned \right.
$$

It follows from relation~(5.4a) and the definition of the function
$h_{\gamma,5}$ (with the help of the definition of the functions
$h_{\gamma,1}$ and $h_{\gamma,3}$) that
$h_{\gamma,5}\in\hat{\bar{\Cal H}}_G^n$, and it is adapted to the
regular system~$\Cal D$. Then relations~(5.3) and the definition of
$h_{\gamma,5}$ also imply that
$\Sigma^\gamma_1=(n+m-2|\gamma|)!I(h_{\gamma,5})$.

On the other hand, I claim that the function $h_\gamma$ defined
in~(5.1) satisfies the identity $h_\gamma=h_{\gamma,6}$. This
statement must be formulated in a more precise form, because the
definition of the function $h_\gamma$ is not unique, we have some
freedom in choosing the indices of its variables. I shall define
such a version of~$h_\gamma$ which provides an appropriate
enumeration of its variables, and the identity
$h_\gamma=h_{\gamma,6}$ holds for this version.

In the definition of $h_\gamma$ we shall work with the function
$\hat h_2(x_{n+1},\dots,x_{n+m})
=h_2(x_{\delta(1)},\dots,x_{\delta(m)})$, with
$\delta(l)=l-n$, $n+1\le l\le n+m$, i.e. we work with a function
with arguments $x_{l+n}$ instead of arguments $x_l$, $1\le l\le m$.
We also replace the set~$A_2$ by its shift~$\bar A_2$ defined as
$\bar A_2=A_2+n=\{\bar r_1,\dots,\bar r_{m-|\gamma|}\}$ with
$\bar r_j=r_j+n$, $1\le j\le m-|\gamma|$, where
$1\le r_1<r_2<\cdots<r_{m-|\gamma|}\le m$ are the elements of the
set~$A_2$. We define the following two functions $\e(\cdot)$ and
$\eta(\cdot)$ on the set $\{n+1,\dots,n+m\}$. We put $\e(l)=1$ if
$l\in\bar A_2$, and $\e(l)=-1$ if
$l\in\{n+1,\dots,n+m\}\setminus A_2$, and we also introduce the
function~$\eta(\cdot)$ defined as $\eta(l)=l$ if $l\in\bar A_2$
and $\eta(l)=p$ with that number~$p$, $1\le p\le n$, for which
$(p,l-n)\in B$ if $l\in\{n+1,\dots,n+m\}\setminus\bar A_2$. With
the help of these functions we define the function
$$
\align
&\hat h_\gamma(x_l, l\in A_1\cup\bar A_2)=\int h_1(x_1,\dots,x_n) \\
&\qquad\qquad\hat h_2(\e(n+1)x_{\eta(n+1)},\dots,\e(n+m)x_{\eta(n+m)})
\prod_{l\in\{1,\dots,n\}\setminus A_1} G(\,dx_l),
\endalign
$$
and define the function $h_\gamma$ by introducing the `right' enumeration
of the variables of the function~$\hat h_\gamma$. For this goal we
define (similarly to the maps~$\beta_1$ and~$\beta_2$ defined before)
the map $\bar\beta\colon\; A_1\cup\bar A_2\to\{1,\dots,n+m-2|\gamma|\}$
as $\bar\beta(p_l)=l$ for $1\le l\le n-|\gamma|$, and
$\bar\beta(\bar r_l)=l+n-|\gamma|$ for $1\le l\le m-|\gamma|$, where
$A_1=\{p_1,\dots,p_{n-|\gamma|}$ with
$1\le p_1<\cdots<p_{n_|\gamma|}\le n$, and
$\bar A_2=A_2+n=\{\bar r_1,\dots,\bar r_{m-|\gamma|}\}$ with
$n+1\le \bar r_1<\cdots<\bar r_{m-|\gamma|}\le n+m$. Then we define
$$
h_\gamma(x_1,\dots,x_{n+m-2|\gamma|})
=\hat h_\gamma(x_{\bar\beta(l)},\;l\in A_1\cup\bar A_2).
$$

To see that $h_{\gamma,6}=h_\gamma$ with the above defined
function~$h_\gamma$ let us first observe that
$$
h_{\gamma,6}(x_1,\dots,x_{n+m-2|\gamma|})=
h_{\gamma,2}(j_{\beta_1(r_1)},\dots,j_{\beta_1(n-|\gamma|))},
k_{\beta_2(1)},\dots,k_{\beta_2(m-|\gamma|)}),
$$
if $x_l\in\Delta_{j_l}$ for all $1\le n+m-2|\gamma|$. On the
other hand, we get, since both functions~$h_1$ and~$h_2$ are
adapted to~$\Cal D$, by applying the definition of the
functions~$h_\gamma$ and~$\hat h_\gamma$ and relation~(5.7)
together with a comparison of the function~$h_2$
with~$\hat h_2$ and of the pair of maps~$\beta_1$
and~$\beta_2$ with the map~$\bar\beta$ that
$$
\align
h_\gamma(x_1,\dots,x_{n+m-2|\gamma|})
&=h_\gamma(u_{j_1},\dots,u_{j_{n+m-2|\gamma|}})
=\hat h_\gamma(u_{\bar\beta(l)},\;l\in A_1\cup\bar A_2)\\
&h_{\gamma,2}(j_{\beta_1(r_1)},\dots,j_{\beta_1(n-|\gamma|))},
k_{\beta_2(1)},\dots,k_{\beta_2(m-|\gamma|)})
\endalign
$$
if $x_l\in\Delta_{j_l}$ for all $1\le n+m-2|\gamma|$.
 
By these identities $h_{\gamma,6}(x_1,\dots,x_{n+m-2|\gamma|})$
and $h_\gamma(x_1,\dots,x_{n+m-2|\gamma|})$ agree in such points
$(x_1,\dots,x_{n+m-2|\gamma})$ for which $x_l\in\Delta_{j_l}$
with some $j_l\in\{\pm1,\dots,\pm N\}$ for all
$1\le l\le n+m-2|\gamma|$. Since both functions $h_{\gamma,6}$
equal zero in other points, this implies that
$h_{\gamma,6}=h_\gamma$, as claimed.

Observe that the function $h_\gamma$ disappears also in such points
$(x_1,\dots,x_{n+m-2|\gamma|})$ for which $x_l\in\Delta_{j_l}$ for
all $1\le l\le n+m-2|\gamma|$ with such indices~$j_l$ for which
some of the numbers in the set $\{\pm j_1,\dots,\pm j_{n-|\gamma|}\}$
or in the set $\{\pm j_{n-|\gamma|+1},\dots,\pm j_{n+m-2|\gamma|}\}$
agree. This fact together with the identity $h_\gamma=h_{\gamma,6}$
and the relation between the functions~$h_{\gamma,5}$
and~$h_{\gamma,6}$ yield the identity
$$
h_\gamma(x_1,\dots,x_{n+m-2|\gamma|})=
h_{\gamma,5}(x_1,\dots,x_{n+m-2|\gamma|})
+h_{\gamma,7}(x_1,\dots,x_{n+m-2|\gamma|})
$$
with
$$
\align
&h_{\gamma,7}(x_1,\dots,x_{n+m-2|\gamma|}) \\
&\qquad =\left\{
\aligned
&h_\gamma(x_1,\dots,x_{n+m-2|\gamma|}) \quad \text{ if there exist
indices }j_l, \;1\le |j_l|\le N,\\
&\qquad 1\le l\le n+m-2|\gamma| \text{ such that } x_l\in \Delta_{j_l},
\; 1\le l\le n+m-2|\gamma|,\\
&\qquad\text{ all numbers } \pm j_1,\dots,\pm j_{n-2|\gamma|}
\text{ are different,}\\
&\qquad \text{ all numbers }\pm j_{n-|\gamma|+1},\dots,\pm j_{n+m-2|\gamma|}
\text{ are different,} \\
&\qquad \text{ and }
\{\pm j_1,\dots,\pm j_{n-|\gamma|}\}\cap\{\pm j_{n-|\gamma|+1},\dots,
\pm j_{n+m-2|\gamma|}\}\neq\emptyset \\
&0 \quad \text{otherwise.}
\endaligned \right.
\endalign
$$

Since $\Sigma^\gamma_1=(n+m-2|\gamma|)!I_G(h_{\gamma,5})$, we have
$$
(n+m-2|\gamma|)!I_G(h_{\gamma})-\Sigma^\gamma_1
=(n+m-2|\gamma|)!I_G(h_{\gamma,7}),
$$
and
$$
E(\Sigma^\gamma_1-(n+m-2|\gamma|)!I_G(h_\gamma))^2
\le(n+m-2|\gamma|)!\|(h_{\gamma,7})\|^2
$$
with the norm $\|\cdot\|$ in $\bar{\Cal H}_G^{n+m-2|\gamma|}$.

On the other hand,
$$
\sup |h_{\gamma,7}(x_1,\dots,x_{n+m-2|\gamma|})|
\le\sup |h_\gamma(x_1,\dots,x_{n+m-2|\gamma|})|
\le K_1 K_2 L^{|\gamma|},
$$
with $K_1=\sup|h_1|$, $K_2=\sup |h_2|$, and
$L=G(A)$, where $A$ is a fixed cube containing all $\Delta_j$. Hence
$$
\aligned
E(\Sigma^\gamma_1-(n+m-2|\gamma|)!I_G(h_\gamma))^2
&\le C_1\|(h_{\gamma,7})\|^2
\le C_2{\sum}''G(\Delta_{j_1})\cdots G(\Delta_{j_{n+m-2|\gamma|}})\\
&\le C\sup_j G(\Delta_j)\le C\e,
\endaligned \tag5.8
$$
where the summation $\sum''$ goes for such sequences
$j_1,\dots,j_{n+m-2|\gamma|}$, $1\le |j_l|\le N$ for all
$1\le l\le n+m-2|\gamma|$,  for which all numbers
$\pm j_1,\dots,\pm j_{n-|\gamma|}$ are different, the same relation
holds for the elements of the sequence
$\pm j_{n-|\gamma|+1},\dots,\pm j_{n+m-2|\gamma|}$, and
$$
\{\pm j_1,\dots,\pm j_{n-|\gamma|}\}\cap
\{\pm j_{n-|\gamma|+1},\dots,\pm j_{n+m-2|\gamma|}\}\neq\emptyset.
$$
The constants $C_1$, $C_2$ and $C$ may depend on the functions~$h_1$,
$h_2$ and spectral measure~$G$, but they do not depend on the regular
system~$\Cal D$, hence in particular on the parameter~$\e$. In the
verification of~(5.8) we can exploit that each term in the sum
$\sum''$ is a product which contains a
factor~$G(\Delta_j)^2\le\e G(\Delta_j)$. Here an argument can be
applied which is similar to the closing step in the proof of
{\it Statement~B}\/ in the proof of the fact that
$\hat{\bar{\Cal H}}_G^n$ is dense in the space~$\bar{\Cal H}_G^n$.

\medskip
Now we turn to the estimation of $E(\Sigma^\gamma_2)^2$. It can be
expressed as a linear combination of terms of the form
$$
\aligned
&\Sigma^\gamma_3(j_p,k_r,j_{\bar p},k_{\bar r},\;p,\bar p\in\{1,\dots,n\},
\; r,\bar r\in\{1,\dots,m\}) \\
&\qquad =E\Biggl(\(\prod_{p\in A_1}Z_G(\Delta_{j_p})
\prod_{r\in A_2}Z_G(\Delta_{k_r})
\prod_{\bar p\in A_1}Z_G(\Delta_{j_{\bar p}})
\prod_{\bar r\in A_2}Z_G(\Delta_{k_{\bar r}})\)\\
&\qquad\qquad\qquad\[\prod_{(p,r)\in B} Z_G(\Delta_{j_p})Z_G(\Delta_{k_r})
-E\prod_{(p,r)\in B} Z_G(\Delta_{j_p})Z_G(\Delta_{k_r})\]\\
&\qquad\qquad\qquad\[\prod_{(\bar p,\bar r)\in B} Z_G(\Delta_{j_{\bar p}})
Z_G(\Delta_{k_{\bar r}})
-E\prod_{(\bar p,\bar r)\in B} Z_G(\Delta_{j_{\bar p}})
Z_G(\Delta_{k_{\bar r}})\]\Biggr),
\endaligned \tag5.9
$$
where $\Sigma^\gamma_3$ depends on such sequences of numbers
$j_p,\,k_r,\,j_{\bar p},\,k_{\bar r}$ with indices $1\le p,\bar p\le n$
and $1\le r,\bar r\le m$  for which
$j_p,k_r,j_{\bar p},k_{\bar r}\in\{\pm1,\dots,\pm N\}$ for all indices
$p,r,\bar p$ and $\bar r$,  $j_p=k_r$ or $j_p=-k_r$ if
$(p,r)\in B$, otherwise all numbers $\pm j_p$, $\pm k_r$ are different,
and the same relations hold for the indices~$j_{\bar p}$
and~$k_{\bar r}$ if $p$ is replaced by $\bar p$ and $r$ is replaced
by~$\bar r$. Moreover the absolute value of all coefficients in this
linear combination is bounded by $\sup|h_1(x)|^2 \sup |h_2(x)|^2$.

We want to show that for most sets of arguments
$(j_p,\,k_r,\,j_{\bar p},\,k_{\bar r})$  $\Sigma^\gamma_3$ equals
zero, and it is also small in the remaining cases.

Let us fix a sequence of arguments $j_p,\,k_r,\,j_{\bar p},\,k_{\bar r}$
of $\Sigma^\gamma_3$, and let us estimate its value with these arguments.
Define the sets
$$
\Cal A=\{j_p\colon\; p\in A_1\}\cup \{k_r\colon\; r\in A_2\}
\text{ \ \ and \ \ }
\bar{\Cal A}=\{j_{\bar p}\colon\; \bar p\in A_1\}\cup
\{k_{\bar r}\colon\; \bar r\in A_2\}.
$$
We claim that $\Sigma^\gamma_3$ equals zero if
$\bar{\Cal A}\neq-\Cal A$. In this case there exists an index
$l\in\Cal A$ such that $-l\notin\bar{\Cal A}$. Let us carry out
the multiplication in~(5.9). Because of the independence
properties of random spectral measures each product in this
expression can be written as the product of independent factors,
and the independent factor containing the term $Z_G(\Delta_l)$ has
zero expectation. To see this observe that the set $\Delta_l$
appears exactly once among the arguments of the terms
$Z_G(\Delta_{j_p})$ and $Z_G(\Delta_{k_r})$, and none of these
terms contains the argument $-\Delta_l=\Delta_{-l}$. Although
$-l\notin\bar{\Cal A}$, it may happen that $l\in\bar{\Cal A}$. In
this case the product under investigation contains the independent
factor $Z_G(\Delta_l)^2$ with $EZ_G(\Delta_l)^2=0$. If
$l\notin\bar{\Cal A}$, then there are two possibilities. Either this
product contains an independent factor of the form $Z_G(\Delta_l)$
with $EZ_G(\Delta_l)=0$, or there is a pair $(\bar p,\bar r)\in B$
such that $(j_{\bar p},k_{\bar r})=(\pm l,\pm l)$, and an
independent factor of the form
$Z_G(\Delta_l)Z_G(\pm\Delta_{-l})Z_G(\pm\Delta_l)$ with the property
$EZ_G(\Delta_l)Z_G(\pm\Delta_{-l})Z_G(\pm\Delta_l)=0$ appears.

Let
$$
\Cal F=\bigcup_{(p,r)\in B}\{j_p,k_r\} \text{ \ \ and \ \ }
\bar{\Cal F}=\bigcup_{(\bar p,\bar r)\in B}\{(j_{\bar p},k_{\bar r}\}.
$$
A factorization argument shows again that the expression in~(5.9)
equals zero if the sets $\Cal F\cup(-\Cal F)$ and
$\bar{\Cal F}\cup(-\bar{\Cal F})$ are  disjoint. We can restrict
ourselves to the case $\Cal A=-\bar{\Cal A}$, and in this case
$\pm\Cal A$ is disjoint both of $\Cal F\cup(-\Cal F)$ and
$\bar{\Cal F}\cup(-\bar{\Cal F})$, and the product under investigation
contains the independent factor
$\prodd_{(p,r)\in B} Z_G(\Delta_{j_p})Z_G(\Delta_{k_r})
-E\prodd_{(p,r)\in B} Z_G(\Delta_{j_p})Z_G(\Delta_{k_r})$ with
expectation zero.

Moreover, if $\Cal F\cup(-\Cal F)$ and
$\bar{\Cal F}\cup(-\bar{\Cal F})$ are not  disjoint, (and
$\Cal A=-\bar{\Cal A}$), then the absolute value of the expression
in~(5.9) can be estimated from above by
$$
C\,\e \prod G(\Delta_{j_p}) G(\Delta_{k_r}) G(\Delta_{j_{\bar p}})
G(\Delta_{k_{\bar r}}) \tag5.10
$$
with a universal constant $C<\infty$ depending only on the parameters~$n$
and~$m$, where the indices $j_p,\,k_r,\,j_{\bar p},\,k_{\bar r}$
are the same as in~(5.9) with
the following difference: All indices appear in~(5.10) with
multiplicity~1, and if both indices $l$ and $-l$ are present
in~(5.9), then one of them is omitted form~(5.10). The multiplying
term~$\e$ appears in~(5.10), since by carrying out the
multiplications in~(5.9) and factorizing each term, we get that all
non-zero terms have a factor
$EZ_G(\Delta)^2Z_G(-\Delta)^2=E(\,\Re Z_G(\Delta)^2+\Im Z_G(\Delta)^2)^2
=E\,\Re Z_G(\Delta)^4+E\,\Im Z_G(\Delta)^4
+2E\,\Re Z_G(\Delta)^2E\,\Im Z_G(\Delta)^2=8G(\Delta)^2$
or $\(E|Z_G(\Delta)|^2\)^2=G(\Delta)^2$, and $G(\Delta)<\e$ for
$\Delta\in\Cal D$. (We did not mention the possibility of an
independent factor of the form $EZ_G(\Delta)^4$ or
$EZ_G(\Delta)^3Z_G(-\Delta)$ with $\Delta\in\Cal D$, because as
some calculation shows, $EZ_G(\Delta)^4=0$ and
$EZ_G(\Delta)^3Z_G(-\Delta)=0$.)

Let us express $E(\Sigma_2^\gamma)^2$ as the linear combination of
the quantities~$\Sigma^\gamma_3$, and let us bound each term
$\Sigma^\gamma_3$ in the above way. This supplies an upper bound
for $E(\Sigma_2^\gamma)^2$ by means of a sum of terms of the
form~(5.10). Moreover, each of these terms appears only with a
multiplicity less than~$C(n,m)$ with an appropriate
constant~$C(n,m)$. Hence we can write
$$
E(\Sigma_2^\gamma)^2\le K_1^2K_2^2 C(n,m)C\e\sum_{r=1}^{n+m}
{\sum_{j_1,\dots,j_r}\!\!}''' G(\Delta_{j_1})\cdots G(\Delta_{j_r}),
$$
where the indices $j_1,\dots,j_r\in\{\pm1,\dots,\pm N\}$ in the
sum $\sum'''$ are all different, and $K_j=\sup |h_j(x)|$, $j=1,2$.
Hence
$$
E(\Sigma_2^\gamma)^2\le C_1\e \sum_{r=1}^{n+m} G(A)^r\le C_2\e
$$
with some appropriate constants $C_1$ and $C_2$. Because of
the inequality~(5.8), the identity
$n!I_G(h_1)m!I_G(h_2)=\summ_{\gamma\in\Gamma}
(\Sigma^\gamma_1+\Sigma_\gamma^2)$ and the last relation one has
$$ \allowdisplaybreaks
\align
&E\(n!I_G(h_1)m!I_G(h_2)-\sum_{\gamma\in\Gamma}(n+m-2|\gamma|)!
I_G(h_\gamma)\)^2\\
&\qquad= E\(\sum_{\gamma\in\Gamma}
\(\Sigma^\gamma_1+\Sigma^\gamma_2-(n+m-2|\gamma|)!\,I_G(h_\gamma)\)\)^2\\
&\qquad \le C_3\(\sum_{\gamma\in\Gamma}
E((m+n-2|\gamma|)!\,I_G(h_\gamma)-\Sigma_1^\gamma)^2
+E(\Sigma_2^\gamma)^2\)\le C_4\e.
\endalign
$$
Since $\e>0$ can be chosen arbitrary small, part~B is proved in the
special case $h_1\in\hat{\bar{\Cal H}}_G^n$,
$h_2\in\hat{\bar{\Cal H}}_G^m$.

If $h_1\in\bar{\Cal H}_G^n$ and $h_2\in\bar{\Cal H}_G^m$, then let
us choose a sequence of functions $h_{1,r}\in\hat{\bar{\Cal H}}_G^n$
and $h_{2,r}\in\hat{\bar{\Cal H}}_G^m$ such that $h_{1,r}\to h_1$
and $h_{2,r}\to h_2$ in the norm of the spaces $\bar{\Cal H}_G^n$
and $\bar{\Cal H}_G^m$ respectively. Define the functions
$\hat h_\gamma(r)$ and $h_\gamma(r)$ in the same way as $h_\gamma$,
but substitute the pair of functions $(h_1,h_2)$ by $(h_{1,r},h_2)$
and $(h_{1,r},h_{2,r})$ in their definition. We shall show by the
help of Part~A) that
$$
E|I_G(h_1)I_G(h_2)-I_G(h_{1,r})I_G(h_{2,r})|\to0,
$$
and
$$
E|I_G(h_\gamma)-I_G(h_\gamma(r))|\to0
\quad\text{for all }\gamma\in\Gamma
$$
as $r\to\infty$. Then a simple limiting procedure shows that
Theorem~5.3 holds for all $h_1\in\bar{\Cal H}_G^n$ and
$h_2\in\bar{\Cal H}_G^m$.

We have
$$
\align
&E|I_G(h_1)I_G(h_2)-I_G(h_{1,r})I_G(h_{2,r})|\\
&\qquad \le E|(I_G(h_1-h_{1,r}))I_G(h_2)|+E|I_G(h_{1,r})I_G(h_2-h_{2,r})|\\
&\qquad \le\frac1{n!\,m!}\(\|h_1-h_{1,r}\|^{1/2}\|h_2\|^{1/2}
+\|h_2-h_{2,r}\|^{1/2}\|h_{1,r}\|\)\to0,
\endalign
$$
and by part~A) of Theorem~5.3
$$
\align
&E|I_G(h_\gamma)-I_G(h_\gamma(r))|\le
E|I_G(h_\gamma)-I_G(\hat h_\gamma(r))|+
E|I_G(h_\gamma(r))-I_G(\hat h_\gamma(r))|\\
&\qquad \le\|h_\gamma-\hat h_\gamma(r)\|^{1/2}
+\|h_\gamma(r)-\hat h_\gamma(r)\|^{1/2} \\
&\qquad \le\|h_1-\hat h_{1,r}\|^{1/2}\|h_2\|^{1/2}
+\|h_2-\hat h_{2,r}\|^{1/2}\|h_{1,r}\|^{1/2}\to0.
\endalign
$$
Theorem~5.3 is proved.

We formulate some consequences of Theorem~5.3. Let
$\bar\Gamma\subset\Gamma$ denote the set of complete diagrams, i.e.\
let a diagram $\gamma\in\bar\Gamma$  if an edge enters in each
vertex of~$\gamma$. We have $EI(h_\gamma)=0$ for all
$\gamma\in\Gamma\setminus\bar\Gamma$, since~(4.3) holds for all
$f\in\bar{\Cal H}_G^n$, $n\ge1$. If $\gamma\in\bar\Gamma$, then
$I(h_\gamma)\in\bar{\Cal H}_G^0$. Let $h_\gamma$ denote the value of
$I(h_\gamma)$ in this case. Now we have the following

\medskip\noindent
{\bf Corollary 5.4.} {\it For all $h_1\in\bar{\Cal H}_G^{n_1}$,\dots,
$h_n\in\bar{\Cal H}_G^{n_m}$
$$
En_1!I_G(h_1)\cdots n_m!I_G(h_m)=
\sum_{\gamma\in\bar\Gamma}h_\gamma.
$$
(The sum on the right-hand side equals zero if $\bar\Gamma$ is
empty.)}

\medskip
As a consequence of Corollary~5.4 we can calculate the expectation of
products of Wick polynomials of Gaussian random variables.

Let $X_{k,j}$, $EX_{k,j}=0$, $1\le k\le p$, $1\le j\le n_k$, be a
sequence of Gaussian random variables. We want to calculate the
expected value of the Wick polynomials
$\:\!X_{k,1}\cdots X_{k,n_k}\!\:$,
\ $1\le k\le p$, if we know all covariances
$EX_{k,j}X_{\bar k,\bar \jmath}=a((k,j),(\bar k,\bar \jmath))$,
$1\le k,\bar k,\le p$, $1\le j\le n_k$, $1\le \bar \jmath\le\bar n_k$.
For this goal let us consider the class of closed diagrams
$\bar\Gamma(k_1,\dots,k_p)$, and define the following quantity
$\gamma(A)$ depending on the closed diagrams $\gamma$ and the set
$A$ of all covariances $EX_{k,j}X_{\bar k,\bar \jmath}=a((k,j),
(\bar k,\bar \jmath))$.
$$
\gamma(A)=\prod_{((k,j),(\bar k,\bar \jmath))\text{ is an edge of }\gamma}
a((k,j),(\bar k,\bar \jmath)), \quad \gamma\in\Gamma.
$$
With the above notation we can formulate the following result.

\medskip\noindent
{\bf Corollary 5.5.} {\it
Let $X_{k,j}$, $EX_{k,j}=0$, $1\le k\le p$, $1\le j\le n_k$, be a
sequence of Gaussian random variables. Let
$a((k,j),(\bar k,\bar \jmath))=EX_{k,j}X_{\bar k,\bar \jmath}$,
$1\le k,\bar k,\le p$, $1\le j\le n_k$, $1\le \bar \jmath\le\bar n_k$
denote the covariances of these random variables. Then
the expected value of the Wick polynomials
$\:\!X_{k,1}\cdots X_{k,n_k}\!\:$, $1\le k\le p$, can be expressed as
$$
E\(\prod_{k=1}^p\:\!X_{k,1}\cdots X_{k,n_k}\!\:\)
=\sum_{\gamma\in\bar\Gamma(k_1,\dots,k_p)} \gamma(A)
$$
with the above defined quantities $\gamma(A)$. In the case when
$\bar\Gamma(k_1,\dots,k_p)$ is empty, e.g. if $k_1+\cdots+k_p$ is an
odd number, the above expectation equals zero.}

\medskip\noindent
{\it Remark.} In the special case when $X_{k,1}=\cdots=X_{k,n_k}=X_k$,
and $EX_k^2=1$ for all indices $1\le k\le p$ Corollary~5.5 provides
a formula for the expectation of the product of Hermite
polynomials of standard normal random variables. In this
case we have $a((k,j),(\bar k,\bar \jmath))=\bar a(k,\bar k)$ with a
function $\bar a(\cdot,\cdot)$ not depending on the arguments~$j$
and~$\bar \jmath$, and the left-hand side of the identity in Corollary~5.5
equals $EH_{n_1}(X_1)\cdots H_{n_p}(X_p)$ with standard normal random
variables $X_1,\dots,X_n$ with correlations
$EX_kX_{\bar k}=\bar a(k,\bar k)$.

\medskip\noindent
{\it Proof of Corollary 5.5.}\/ We can represent the random variables
$X_{k,j}$ in the form $X_{k,j}=\summ_pc_{k,j,p}\xi_p$ with some
appropriate coefficients $c_{k,j,p}$, where $\xi_1,\xi_2,\dots$ is
a sequence of independent standard normal random variables. Let
$Z(\,dx)$ denote a random spectral measure corresponding to the
one-dimensional spectral measure with density function
$g(x)=\frac1{2\pi}$ for $|x|<\pi$, and $g(x)=0$ for $|x|\ge\pi$. The
random integrals $\int e^{ipx}Z(\,dx)$, $p=0,\pm1,\pm2,\dots$, are
independent standard normal random variables.
Define $h_{k,j}(x)=\summ_p c_{k,j,p}e^{ipx}$, $k=1,\dots,p$,
$1\le j\le n_k$. The random variables $X_{k,j}$ can be identified
with the random integrals $\int h_{k,j}(x)Z(\,dx)$, $k=1,\dots,p$,
$1\le j\le n_k$, since their joint distributions coincide. Put
$\hat h_k(x_1,\dots,x_{n_k})=\prodd_{j=1}^{n_k}h_{k,j}(x_j)$.
It follows from Theorem~4.6 that
$$
\:\!X_{k,1}\cdots X_{k,n_k}\!\:=
\int\hat h_k(x_1,\dots,x_{n_k}) Z(\,dx_1)\dots Z(\,dx_{n_k})
=n_k!I(\hat h_k(x_1,\dots,x_{n_k}))
$$
for all $1\le k\le p$. Hence an application of Corollary~5.4
yields Corollary~5.5. One only has to observe that
$\int_{-\pi}^\pi h_{k,j}(x)\overline{h_{\bar k,\bar \jmath}(x)}\,dx
=a((k,j),(\bar k,\bar \jmath))$ for all $k,\,k=1,\dots,p$ and
$1\le j\le n_k$.

\medskip
Theorem~5.3 states in particular that the product of Wiener--It\^o
integrals with respect to a random spectral measure  of a stationary
Gaussian fields belongs to the Hilbert space $\Cal H$ defined by this
field, since it can be written as a sum of Wiener--It\^o integrals.
This means a trivial measurability condition, and also that the
product has a finite second moment, which is not so trivial.
Theorem~5.3 actually gives the following non-trivial inequality.

Let $h_1\in\Cal H_G^{n_1}$,\dots, $h_m\in\Cal H_G^{n_m}$. Let
$|\bar\Gamma(n_1,n_1,\dots,n_m,n_m)|$ denote the number of complete
diagrams in $\bar\Gamma(n_1,n_1,\dots,n_m,n_m)$, and put
$$
C(n_1,\dots,n_m)=\frac{|\bar\Gamma(n_1,n_1,\dots,n_m,n_m)|}
{n_1!\cdots n_m!}.
$$
In the special case $n_1=\cdots=n_m=n$ let
$\bar C(n,m)=C(n_1,\dots,n_m)$. Then

\medskip\noindent
{\bf Corollary 5.6.} {\it
$$
E\[(n_1!I_G(h_1))^2\cdots (n_m! I_G(h_m))^2\]\le C(n_1,\dots,n_m)
E(n_1!I_G(h_1))^2
\cdots (n_m! E(I_G(h_m))^2.
$$
In particular,
$$
E\[(n!I_G(h))^{2m}\]\le\bar C(n,m)(E(n!I_G(h))^2)^m\quad \text{ if \ }
h\in \Cal H_G^n.
$$
}

\medskip\noindent
Corollary~5.6 follows immediately from Corollary~5.4 by applying it
first for the sequence $h_1,h_1,\dots,h_m,h_m$ and then for the pair
$h_j,h_j$ which yields that $E (n_j!I_G(h_j))^2=n_j!\|h_j\|^2$,
$1\le j\le m$. One only has to observe that
$|h_\gamma|\le\|h_1\|^2\cdots\|h_m\|^2$ for all complete diagrams
by Part~A) of Theorem~5.3.

The inequality in Corollary~5.6 is sharp. If $G$ is a finite
measure and $h_1\in H_G^{n_1}$,\dots, $h_m\in H^{n_m}_G$ are constant
functions, then equality can be written in Corollary~5.6. We remark
that in this case $I_G(h_1),\dots,I_G(h_m)$ are constant times the
$n_1$-th,\dots, $n_m$-th Hermite polynomials of the same standard
normal random variable. Let us emphasize that the constant
$C(n_1,\dots,n_m)$ depends only on the parameters $n_1,\dots,n_m$
and not on the form of the functions $h_1,\dots,h_m$. The function
$C(n_1,\dots,n_m)$ is monotone in its arguments. The following
argument shows that
$$
{C(n_1+1,n_2,\dots,n_m)}\ge{C(n_1,\dots,n_m)}
$$

Let us say that two complete diagrams in
$\bar\Gamma(n_1,n_1,\dots,n_m,n_m)$ or in
$\bar\Gamma(n_1+1,n_1+1,\dots,n_m,n_m)$ are equivalent if they
can be transformed into each other by permuting the vertices
$(1,1),\dots,(1,n_1)$ in $\bar\Gamma(n_1,n_1,\dots,n_m,n_m)$ or
the vertices $(1,1),\dots,(1,n_1+1)$ in
$\bar\Gamma(n_1+1,n_1+1,\dots,n_m,n_m)$. The equivalence classes
have $n_1!$ elements in the first case and $(n_1+1)!$ elements
in the second one. Moreover, the number of equivalence classes is
less in the first case than in the second one. (They would agree if
we counted only those equivalence classes in the second case which
contain a diagram where $(1,n_1+1)$ and $(2,n_1,1)$ are connected
by an edge. Hence
$$
\frac1{n_1!}|\bar\Gamma(n_1,n_1,\dots,n_m,n_m)|\le
\frac1{(n_1+1)!}|\bar\Gamma(n_1+1,n_1+1,\dots,n_m,n_m)|
$$
as we claimed.

The next result, formulated in a more elementary way, may better
illuminate the content of Corollary~5.6.

\medskip\noindent
{\bf Corollary 5.7.} {\it Let $\xi_1,\dots,\xi_k$ be a normal random
vector, and $P(x_1,\dots,x_k)$ a polynomial of degree~$n$. Then
$$
E\[P(\xi_1,\dots,\xi_k)^{2m}\]\le\bar C(n,m)(n+1)^m
\(EP(\xi_1,\dots,\xi_k)^2\)^m.
$$
}

\medskip
The multiplying constant $\bar C(n,m)(n+1)^m$ is not sharp in this
case. Observe that it does not depend on~$k$.

\medskip\noindent
{\it Proof of Corollary 5.7.}\/ We can write $\xi_j=\int f_j(x)Z(\,dx)$
with some $f_j\in\Cal H^1$, $j=1,2,\dots,k$, where $Z(\,dx)$ is
the same as in the proof of Corollary~5.5. There exist some
$h_j\in\Cal H^j$, $j=0,1,\dots,n$, such that
$$
P(\xi_1,\dots,\xi_k)=\sum_{j=0}^n j! I(h_j).
$$
Then
$$
\align
&EP(\xi_1,\dots,\xi_k)^{2m}=E\[\(\sum_{j=0}^nj!I(h_j)\)^{2m}\]
\le (n+1)^m E\[\sum_{j=0}^n (j!I(h_j))^2\]^m   \\
&\qquad\le (n+1)^m\sum_{p_1+\cdots+p_n=m}C(p_1,\dots,p_n)(EI(h_0)^2)^{p_0}
\cdots (En!I(h_n)^2)^{p_n}\frac{m!}{p_1!\cdots p_n!} \\
&\qquad\le (n+1)^m\bar C(n,m)\sum_{p_1+\cdots+p_n=m} (EI(h_0)^2)^{p_0}
\cdots (EI(n!h_n)^2)^{p_n}\frac{m!}{p_1!\cdots p_n!} \\
&\qquad=(n+1)^m\bar C(n,m)\[\sum E (j!I(h_j))^2\]^m=(n+1)^m \bar C(n,m)
\(EP(\xi_1,\dots,\xi_k)^2\)^m.
\endalign
$$

\beginsection 6. Subordinated random fields. Construction of
self-similar fields.

Let $X_n$, $n\in\text{\BBB Z}_\nu$, be a discrete stationary Gaussian
random field, and let the random field $\xi_n$, $n\in\text{\BBB Z}_\nu$,
be subordinated to it. Let $Z_G$ denote the random spectral measure
adapted to the random field $X_n$. By Theorem~4.1 the random variable
$\xi_0$ can be represented as
$$
\xi_0=f_0+\sum_{k=1}^\infty \frac1{k!}\int
f_k(x_1,\dots,x_k)Z_G(\,dx_1)\dots Z_G(\,dx_k)
$$
with an appropriate $f=(f_0,f_1,\dots)\in\,\text{Exp}\,{\Cal H}_G$
in a unique way. This formula together with Theorem~4.3 yields the
following

\medskip\noindent
{\bf Theorem 6.1.} {\it A random field $\xi_n$, $n\in\text{\BBB Z}_\nu$,
subordinated to the stationary Gaussian random field $X_n$,
$n\in\text{\BBB Z}_\nu$, can be written in the form
$$
\xi_n=f_0+\sum_{k=1}^\infty \frac1{k!}\int e^{i((n,x_1+\cdots+x_k)}
f_k(x_1,\dots,x_k)Z_G(\,dx_1)\dots Z_G(\,dx_k), \quad n\in{\BBB Z}_\nu,
\tag6.1
$$
with some $f=(f_0,f_1,\dots)\in\,\text{\rm Exp}\,{\Cal H}_G$,
where~$G$ is the spectral measure of the field $X_n$, and $Z_G$ is
the random spectral measure adapted to it. This representation is
unique. It is also clear that formula~(6.1) defines a subordinated
field for all $f\in\,\text{\rm Exp}\,{\Cal H}_G$.}

\medskip\noindent
If the spectral measure $G$ has the property $G(\{x\colon\;x_p=u\})=0$
for all $u\in R^1$ and $1\le p\le k$, where $x=(x_1,\dots,x_k)$
(this is a strengthened form of the non-atomic property), then the
functions
$$
\bar f_k(x_1,\dots,x_k)=f_k(x_1,\dots,x_k)\tilde\chi_0^{-1}(x_1+\cdots+x_k),
\quad k=1,2,\dots,
$$
are meaningful, as functions in the measure space
$(R^{k\nu},\Cal B^{k\nu},G^k)$, where
$\tilde\chi_n(x)=e^{i(n,x)}\prodd_{p=0}^\nu\frac{e^{ix^{(p)}}-1}{ix^{(p)}}$,
$n\in\text{\BBB Z}_\nu$, denotes the Fourier transform of the uniform
distribution on the $\nu$-dimensional unit cube
$\prod\limits_{p=1}^\nu[n^{(p)},n^{(p)}+1]$. Then the random
variable $\xi_n$ in formula~(6.1) can be rewritten in the form
$$
\xi_n=f_0+\sum_{k=1}^\infty \frac1{k!}\int\tilde\chi_n(x_1+\cdots+x_k)
\bar f_k(x_1,\dots,x_k)Z_G(\,dx_1)\dots Z_G(\,dx_k).
$$
Hence the following Theorem~$6.1'$ can be considered as the continuous
time version of Theorem~6.1.

\medskip\noindent
{\bf Theorem~$6.1'$.} {\it Let the generalized random field $\xi(\varphi)$,
$\varphi\in\Cal S$, be subordinated to the stationary Gaussian
generalized random field $X(\varphi)$, $\varphi\in\Cal S$. Let $G$
denote the spectral measure of the field $X(\varphi)$, and let $Z_G$
be the random spectral measure adapted to it. Then $\xi(\varphi)$ can
be written in the form
$$
\xi(\varphi)=f_0\cdot\tilde\varphi(0)+\sum_{k=1}^\infty\frac1{k!}\int
\tilde\varphi(x_1+\cdots+x_k)f_k(x_1,\dots,x_k)Z_G(\,dx_1)\dots
Z_G(\,dx_k), \tag$6.1'$
$$
where the functions $f_k$ are invariant under all permutations of their
variables,
$$
f_k(-x_1,\dots,-x_k)=\overline{f_k(x_1,\dots,x_k)}, \quad k=1,2,\dots,
$$
and
$$
\sum_{k=1}^\infty \frac1{k!}\int (1+|x_1+\cdots+x_k|^2)^{-p}
|f_k(x_1+\cdots+x_k)|^2G(\,dx_1)\dots G(\,dx_k)<\infty \tag6.2
$$
with an appropriate number $p>0$. This representation is unique.

Contrariwise, all random fields $\xi(\varphi)$, $\varphi\in\Cal S$,
defined by formulas~$(6.1')$ and~(6.2) are subordinated to the
stationary, Gaussian random field $X(\varphi)$,~$\varphi\in\Cal S$.}

\medskip\noindent
{\it Proof of Theorem~$6.1'$}.\/ It is clear that a random field
$\xi(\varphi)$,~$\varphi\in\Cal S$, defined by~$(6.1')$ and~(6.2)
is subordinated to~$X(\varphi)$. One has to check that the definition
of~$\xi(\varphi)$ in formula~($6.1'$) is meaningful for
all~$\varphi\in\Cal S$, because of~(6.2),
$\xi(T_t\varphi)=T_t\xi(\varphi)$ for all shifts~$T_t$,~$t\in R^\nu$,
by Theorem~4.3, and also the following continuity property holds. For
all $\e>0$ there is a small neighbourhood~$H$ of the origin in the
space~$\Cal S$ such that if $\varphi=\varphi_1-\varphi_2\in H$ for
some $\varphi_1,\varphi_2\in\Cal S$ then
$E[\xi(\varphi_1)-\xi(\varphi_2)]^2=E\xi\varphi)^2<\e^2$.

Since the Fourier transform $\varphi(\cdot)\to\tilde\varphi(\cdot)$
is a bicontinuous map in $\Cal S$, to prove the above continuity
property it is enough to check that $E\xi(\varphi)^2<\e^2$ if
$\tilde\varphi\in H$ for an appropriate small neighbourhood~$H$
of the origin in~$\Cal S$. But this relation holds with the choice
$H=\{\varphi\colon (1+|x|^2)^p|\varphi(x)|\le\frac{\e^2}K
\text{ for all }x\in R^\nu\}$ with a sufficiently large $K>0$
because of condition~(6.2).

To prove that all subordinated fields have the above representation
observe that the relation
$$
\xi(\varphi)=\Psi_{\varphi,0}+\sum_{k=1}^\infty\frac1{k!}\int
\Psi_{\varphi,k}(x_1,\dots,x_k)Z_G(\,dx_1)\dots Z_G(\,dx_k) \tag6.3
$$
holds for all $\varphi\in\Cal S$ with some
$(\Psi_{\varphi,0},\Psi_{\varphi,1},\dots)\in\,\text{Exp}\,\Cal H_G$
depending on the function~$\varphi$. We are going to show that these
functions $\Psi_{\varphi,k}$ can be given in the form
$$
\Psi_{\varphi,k}(x_1,\dots,x_k)=f_k(x_1,\dots,x_k)\cdot\tilde\varphi
(x_1+\cdots+x_k), \quad k=1,2,\dots,
$$
with some functions $f_k\in\Cal B^{k\nu}$, and
$$
\Psi_{\varphi,0}=f_0\cdot\tilde\varphi(0)
$$
for all $\varphi\in\Cal S$ with a sequence of functions
$f_0,f_1,\dots$ not depending on~$\varphi$.

To show this let us choose a $\varphi_0\in\Cal S$ such that
$\tilde\varphi_0(x)>0$ for all $x\in R^{\nu}$. (We can make
for instance the choice $\varphi_0(x)=e^{-(x,x)}$.) We claim
that the finite linear combinations
$\sum a_p\varphi_0(x-t_p)=\sum a_pT_{t_p}\varphi_0(x)$ are
dense in $\Cal S$. To prove this it is enough to show that the
functions $\psi$ whose Fourier transforms $\tilde\psi$ have a
compact support can well be approximated by such linear combinations,
because these functions $\psi$ are dense in $\Cal S$. (The statement
that these functions $\psi$ are dense in $\Cal S$ is equivalent to
the statement that their Fourier transform $\tilde\psi$ are dense
in the space $\tilde{\Cal S}\subset\Cal S^c$ consisting of the
Fourier transforms of the (real valued) functions in the
space~$\Cal S$.) We have
$\frac{\tilde\psi}{\tilde\varphi_0}\in\Cal S^c$ for such
functions~$\psi$, where $\Cal S^c$ denotes the Schwartz-space of
complex valued, at infinity strongly decreasing, smooth functions
again, because $\tilde\varphi_0(x)\neq0$, and $\tilde\psi$ has a
compact support. There exists a function $\chi\in\Cal S$ such that
$\tilde\chi=\frac{\tilde\psi}{\tilde\varphi_0}$. (Here we exploit
that the space of Fourier transforms of the functions from $\Cal S$
agrees with the space of those functions $f\in\Cal S^c$ for which
$f(-x)=\overline{f(x)}$.) Therefore
$\psi(x)=\chi*\varphi_0(x)=\int \chi(t)\varphi_0(x-t)\,dt$, where
$*$ denotes convolution. It can be seen by
exploiting this relation together with the rapid decrease of~$\chi$
and~$\varphi_0$ together of its derivatives at infinity, and
approximating the integral defining the convolution by an appropriate
final sum that for all integers $r>0$, $s>0$ and real numbers~$\e>0$
there exists a finite linear combination
$\hat\psi(x)=\hat\psi_{r,s,\e}(x)=\summ_p a_p\varphi_0(x-t_p)$ such
that $(1+|x|^s)|\psi(x)-\hat\psi(x)|<\e$ for all $x\in R^\nu$, and
the same estimate holds for all derivatives of $\psi(x)-\hat\psi(x)$
of order less than~$r$. Beside this, also the relation
$\hat\psi(-x)=\overline{\hat\psi(x)}$ holds (similarly to the relation
$\psi(-x)=\overline{\psi(x)}$).

I only briefly explain why such an approximation exists. Some
calculation enables us to reduce this statement to the case when
$\psi=\chi*\varphi_0$ with a function $\chi\in\Cal D$, which has
compact support. To give the desired approximation choose a small
number $\delta>0$, introduce the cube
$\Delta=\Delta(\delta)=[-\delta,\delta)^\nu\subset R^\nu$ and
define the vectors $k(\delta)=(2k_1\delta,\dots,2k_\nu\delta)\in R^\nu$
for all $k=(k_1,\dots,k_\nu)\in\text{\BBB Z}_\nu$.
Given a fixed vector $x\in R^\nu$ let us define the vector
$u(x)\in R^\nu$ for all $u\in R^\nu$ as $u(x)=x+k(\delta)$ with that
vector $k\in\text{\BBB Z}_\nu$ for which $x+k(\delta)-u\in\Delta$,
and put $\varphi_{0,x}(u)=\varphi_0(u(x))$. It can be seen that
$\hat\psi(x)=\chi*\varphi_{0,x}(x)$ is a finite linear combination
of numbers of the form $\varphi_0(x-t_k)$ (with $t_k=k(\delta)$)
with coefficients not depending on~$x$. Moreover, if $\delta>0$
is chosen sufficiently small (depending on $r,s$ and~$\e$), then
$\hat\psi(x)=\hat\psi_{r,s,\e}(x)$ has all properties we demanded.

The above argument implies that there is a sequence of functions
$\hat\psi_{r,s,\e}$ which converges to the function~$\psi$ in the
topology of the space~$\Cal S$. As a consequence, the finite linear
combinations $\sum a_p\varphi_0(x-t_p)$ are dense in~$\Cal S$.

Define
$$
f_k(x_1,\dots,x_k)
=\frac{\Psi_{\varphi_0,k}(x_1,\dots,x_k)}{\tilde\varphi_0(x_1+\cdots+x_k)},
\quad k=1,2,\dots,\quad \text{and }
f_0=\frac{\Psi_{\varphi_0,0}}{\tilde\varphi_0(0)}.
$$
If $\varphi(x)=\sum a_p\varphi_0(x-t_p)=\sum a_pT_{t_p}\varphi_0(x)$,
and the sum defining~$\varphi$ is finite, then by Theorem~4.3
$$
\align
\xi(\varphi)&=\(\sum a_p\)f_0\cdot\tilde\varphi_0(0)
+\sum_{k=1}^\infty\frac1{k!}\int
\sum_p a_pe^{i(t_p,x_1+\cdots+x_k)}
\tilde\varphi_0(x_1+\cdots+x_k) \\
&\qquad\qquad\cdot f_k(x_1,\dots,x_k)Z_G(\,dx_1)\dots Z_G(\,dx_k),\\
&=f_0\cdot\tilde\varphi(0)+\sum_{k=1}^\infty\frac1{k!}\int
\tilde\varphi(x_1+\cdots+x_k)
f_k(x_1,\dots,x_k)Z_G(\,dx_1)\dots Z_G(\,dx_k).
\endalign
$$
Relation~(6.3) holds for all $\varphi\in\Cal S$, and there exists
a sequence
$\varphi_j(x)=\sum\limits_p a_p^{(j)}\varphi_0(x-t_p^{(j)})\in\Cal S$
satisfying~$(6.1')$ such that $\varphi_j\to\varphi$ in the topology
of~$\Cal S$. This implies that
$\lim E[\xi(\varphi_j)-\xi(\varphi)]^2\to0$, and in particular
$EI_G(\Psi_{\varphi,k}-\hat\varphi_{j,k}f_k)^2\to0$ with
$\hat\varphi_{j,k}(x_1,\dots,x_k)=\tilde\varphi_j(x_1+\cdots+x_k)$
as $j\to\infty$ for all $k=1,2,\dots$. (To carry out some further
argument we restricted the domain of integration to a bounded
set~$A$.) Hence
$$
\int_A|\Psi_{\varphi,k}(x_1,\dots,x_k)-\tilde\varphi_j(x_1+\cdots+x_k)
f_k(x_1,\dots,x_k)|^2G(\,dx_1)\dots G(\,dx_k)\to0
$$
as $j\to\infty$ for all $k$ and for all bounded sets $A\in R^{k\nu}$.
On the other hand,
$$
\int_A|\tilde\varphi(x_1+\cdots+x_k)-\tilde\varphi_j(x_1+\cdots+x_k)|^2
|f_k(x_1,\cdots,x_k)|^2G(\,dx_1)\dots G(\,dx_k)\to0,
$$
since $\tilde\varphi_j(x)-\tilde\varphi(x)\to0$ in the supremum
norm if $\tilde\varphi_j\to\tilde\varphi$ in the topology of $\Cal S$,
and the property $\tilde\varphi_0(x)>0$ together with the continuity
of $\tilde\varphi_0$ and the inequality
$EI_G(\hat\varphi_{0,k}f_k)^2<\infty$ imply that
$\int_A |f_k(x_1,\dots,x_k)|^2G(\,dx_1)\dots G(dx_k)<\infty$
on all bounded sets~$A$. The last two relations yield that
$$
\Psi_{\varphi,k}(x_1,\dots,x_k)
=\tilde\varphi(x_1+\cdots+x_k)f_k(x_1,\dots,x_k), \quad k=1,2,\dots.
$$
Similarly,
$$
\psi_{\varphi,0}=\tilde\varphi(0)f_0.
$$
These relations imply~$(6.1')$.

To complete the proof of Theorem~$6.1'$ we show that~(6.2) follows
from the continuity of the transformation
$F\colon\;\varphi\to \xi(\varphi)$ from the space $\Cal S$ into the
space $L^2(\Omega,\Cal A,P)$.

We recall that the transformation $\varphi\to\tilde\varphi$ is
bicontinuous in $\Cal S^c$. Hence the transformation
$\tilde\varphi\to\xi(\varphi)$ is a continuous map from the space of
the Fourier transforms of the functions in the space~$\Cal S$ to
$L^2(\Omega,\Cal A,P)$. This continuity implies that there exist
some integers $p>0$, $r>0$ and real number $\delta>0$ such that if
$$
(1+|x^2|)^p\left|\frac{\partial^{s_1+\cdots+s_\nu}}{\partial {x^{(1)}}^{s_1}
\dots\partial {x^{(\nu)}}^{s_\nu}}\tilde\varphi(x)\right|<\delta
\quad\text{for all } s_1+\cdots+s_\nu\le r, \tag6.4
$$
then $E\xi(\varphi)^2\le1$.

Let us choose a $\psi\in\Cal S$ such that $\psi$ has a compact
support, $\psi(x)=\psi(-x)$, $\psi(x)\ge0$ for all $x\in R^\nu$,
and $\psi(x)=1$ if $|x|\le 1$. (There exist such functions.)
Define the functions $\tilde\varphi_m(x)=C(1+|x|^2)^{-p}\psi(\frac xm)$.
Then $\varphi_m\in\Cal S$, since its Fourier transform
$\tilde\varphi_m$ is an even function, and it is in the
space~$\Cal S$ being an infinite many times differentiable function
with compact support. Moreover, $\varphi_m$ satisfies~(6.4)
for all $m=1,2,\dots$ if the number $C>0$ in its definition is
chosen sufficiently small. This number~$C$ can be chosen
independently of~$m$. (To see this observe that $(1+|x^2|)^{-p}$
together with all of its derivatives of order not bigger than~$r$
can be bounded by $\frac{C(p,r)}{(1+|x|^2)^p}$ with an appropriate
constant~$C(p,r)$.) Hence
$$
E\xi(\varphi_m)^2=\sum\frac1{k!}\int|\tilde\varphi_m(x_1+\cdots+x_k)|^2
|f_k(x_1,\cdots,x_k)|^2G(dx_1)\dots G(dx_k)\le1
$$
for all $m=1,2,\dots$.

As $\tilde\varphi_m(x)\to C(|1+|x|^2)^{-p}$ as $m\to\infty$, and
$\tilde\varphi_k(x)\ge0$, an $m\to\infty$ limiting procedure in the
last relation together with Fatou's lemma imply that
$$
C\sum\frac1{k!}\int(1+|x_1+\cdots+x_k)|^2)^{-p}
|f_k(x_1,\cdots,x_k)|^2G(dx_1)\dots G(dx_k)\le1.
$$
Theorem~$6.1'$ is proved.

\medskip
We shall call the representations given in Theorems~6.1 and~$6.1'$
the canonical representation of a subordinated field. From now on
we restrict ourselves to the case $E\xi_n=0$ or $E\xi(\varphi)=0$
respectively, i.e. to the case when $f_0=0$ in the canonical
representation. If
$$
\xi(\varphi)=\sum_{k=1}^\infty\frac1{k!}\int\tilde\varphi(x_1+\cdots+x_k)
f_k(x_1,\dots,x_k)Z_G(\,dx_1)\dots Z_G(\,dx_k),
$$
then
$$
\xi(\varphi^A_t)=\sum_{k=1}^\infty\frac1{k!}\frac{t^\nu}{A(t)}
\int\tilde\varphi(t(x_1+\cdots+x_k))
f_k(x_1,\dots,x_k)Z_G(\,dx_1)\dots Z_G(\,dx_k)
$$
with the function $\varphi^A_t$ defined in~(1.3). Define the spectral
measures $G_t$ by the formula $G_t(A)=G(tA)$. Then we have by Lemma~4.5
$$
\xi(\varphi^A_t)\overset\Delta\to=\sum_{k=1}^\infty\frac1{k!}
\frac{t^\nu}{A(t)}
\int\tilde\varphi(x_1+\cdots+x_k)
f_k\(\frac{x_1}t,\dots,\frac{x_k}t\)Z_{G_t}(\,dx_1)\dots Z_{G_t}(\,dx_k).
$$

If $G(tB)=t^{2\kappa}G(B)$ with some $\kappa>0$ for all $t>0$ and
$B\in\Cal B^\nu$, $f_k(\lambda x_1,\dots,\lambda x_k)=
\lambda^{\nu-\kappa k-\alpha}f_k(x_1,\dots,x_k)$, and $A(t)$ is
chosen as $A(t)=t^\alpha$, then Theorem~4.4 (with the choice
$G'(B)=G(tB)=t^{2\kappa}G(B)$) implies that
$\xi(\varphi_t^A)\overset\Delta\to=\xi(\varphi)$.
Hence we obtain the following

\medskip\noindent
{\bf Theorem 6.2.} {\it Let the generalized random field $\xi(\varphi)$
be given by the formula
$$
\xi(\varphi)=\sum_{k=1}^\infty\frac1{k!}\int \tilde\varphi
(x_1+\cdots+x_k)f_k(x_1,\dots,x_k)Z_G(\,dx_1)\dots Z_G(\,dx_k).
\tag6.5
$$
If $f_k(\lambda x_1,\dots,\lambda x_k)=\lambda^{\nu-\kappa k-\alpha}
f_k(x_1,\dots,x_k)$ for all~$k$, $(x_1,\dots,x_k)\in R^{k\nu}$ and
$\lambda>0$, $G(\lambda A)=\lambda^{2\kappa}G(A)$ for all
$\lambda>0$ and $A\in\Cal B^\nu$, then $\xi$ is a self-similar
random field with parameter $\alpha$.}

\medskip
The discrete time version of this result can be proved in the same
way. It states the following

\medskip\noindent
{\bf Theorem 6.2$'$.} {\it If the discrete random field $\xi_n$,
$n\in\text{\BBB Z}_\nu$, has the form
$$
\xi_n=\sum_{k=1}^\infty \frac1{k!}\int\tilde\chi_n(x_1+\cdots+x_k)
f_k(x_1,\dots,x_k)Z_G(\,dx_1)\dots Z_G(\,dx_k), \quad
n\in\text{\BBB Z}_\nu, \tag$6.5'$,
$$
and $f_k(\lambda x_1,\dots,\lambda x_k)=\lambda^{\nu-\kappa k-\alpha}
f_k(x_1,\dots,x_k)$ for all~$k$, $G(\lambda A)=\lambda^{2\kappa}G(A)$,
then $\xi_n$ is a self-similar random field with parameter~$\alpha$.}

\medskip
Theorems~6.2 and~$6.2'$ enable us to construct self-similar random
fields. Nevertheless, we have to check whether formulas~(6.5)
and~$(6.5')$  are meaningful. The hard part of this problem is
to check whether
$$
\sum \frac1{k!}\int |\tilde\chi_n(x_1+\cdots+x_k)|^2
|f_k(x_1,\dots,x_k)|^2 G(\,dx_1)\dots G(\,dx_k)<\infty,
$$
or whether
$$
\sum \frac1{k!}\int |\tilde\varphi(x_1+\cdots+x_k)|^2
|f_k(x_1,\dots,x_k)|^2 G(\,dx_1)\dots G(\,dx_k)<\infty
\quad \text{for all }\varphi\in\Cal S.
$$
To investigate when these expressions are finite is a rather hard
problem in the general case. The next result enables us to prove
the finiteness of these expressions in some interesting cases.

Let us define the measure~$G$
$$
G(A)=\int_A |x|^{2\kappa-\nu}a\(\frac x{|x|}\)\,dx,\quad A\in\Cal B^\nu,
\tag6.6
$$
where $a(\cdot)$ is a non-negative, measurable and even function on the
$\nu$-dimensional unit sphere $S_{\nu-1}$, and $\kappa>0$. (The
condition $\kappa>0$ is imposed to guarantee the relation $G(A)<\infty$
for all bounded sets $A\in\Cal B^\nu$.) We prove the following
 
\medskip\noindent
{\bf Proposition 6.3.} {\it Let the measure $G$ be the same as in
formula~(6.6).

\medskip
\item{a)} If the function $a(\cdot)$ is bounded on the unit sphere
$S_{\nu-1}$, and $\frac \nu k>2\kappa>0$, then
$$
\align
D(\varphi)&=\int|\tilde\varphi(x_1+\cdots+x_k)|^2
G(\,dx_1)\dots G(\,dx_k)\\
&\le C\int(1+|x_1+\cdots+x_k)|^2)^{-p} G(\,dx_1)\dots G(\,dx_k)<\infty
\endalign
$$
if $\varphi\in\Cal S$ for all $p>\frac\nu2$ with some
$C=C(\varphi,p)<\infty$, and
$$
D(n)=\int|\tilde\chi_n(x_1+\cdots+x_k)|^2 G(\,dx_1)\dots G(\,dx_k)<\infty
\quad \text{for all \ } n\in\text{\BBB Z}_\nu.
$$

\item{b)} If there is a constant $C>0$ such that $a(x)>D$ in a
neighbourhood of a point $x_0\in S_{\nu-1}$, and either $2\kappa\le0$ of
$2\kappa\ge\frac\nu k$, then the integrals $D(n)$ and some $D(\varphi)$,
$\varphi\in\Cal S$, are divergent.

}

\medskip\noindent
{\it Proof of Proposition 6.3.}\/ {\it Proof of Part a)}

\noindent
We may assume that $a(x)=1$ for all $x\in S_{\nu-1}$. Define
$$
J_{\kappa,k}(x)=\int_{x_1+\cdots+x_k=x}
|x_1|^{2\kappa-\nu}\cdots|x_k|^{2\kappa-\nu}
\,dx_1\dots\,dx_k, \quad x\in R^{\nu}
$$
for $k\ge2$, where $\,dx_1\dots\,dx_k$ denotes the Lebesgue
measure on the hyperplane $x_1+\cdots+x_k=x$, and let
$J_{\kappa,1}(x)=|x|^{2\kappa-\nu}$. We have
$$
J_{\kappa,k}(\lambda x)
=|\lambda|^{k(2\kappa-\nu)+(k-1)\nu}J_{\kappa,k}(x),
=|\lambda|^{2k\kappa-\nu}J_{\kappa,k}(x),
\quad x\in R^\nu\;\; \lambda>0,
$$
because of the homogeneity of the integral. Beside this
$$
\aligned
&\qquad\qquad D(n)=\int_{R^\nu}|\tilde\chi_n(x)|^2 J_{\kappa,k}(x)\,dx,\\
&\int (1+|x_1+\cdots+x_k|^2)^{-p}G(\,dx_1)\dots G(\,dx_k)
=\int (1+|x|^2)^{-p} J_{\kappa,k}(x)\,dx.
\endaligned \tag6.7
$$
We prove by induction on $k$ that
$$
J_{\kappa,k}(x)\le C(\kappa,k)|x|^{2\kappa k-\nu} \tag6.8
$$
with an appropriate constant $C(\kappa,k)<\infty$ if
$\frac\nu k>2\kappa>0$.

We have
$$
J_{\kappa,k}(x)=\int J_{\kappa,k-1}(y)|x-y|^{2\kappa-\nu}\,dy.
$$
Hence
$$
\align
J_{\kappa,k}&\le C(\kappa,k-1)\int
|y|^{(2\kappa(k-1)-\nu}|x-y|^{2\kappa-\nu}\,dy\\
&=C(\kappa,k-1)|x|^{2\kappa k-\nu}
\int |y|^{(2\kappa(k-1)-\nu}\left|\frac x{|x|}-y\right|^{2\kappa-\nu}\,dy
=C(\kappa,k)|x|^{2\kappa k-\nu},
\endalign
$$
since $\int |y|^{(2\kappa(k-1)-\nu}
\left|\frac x{|x|}-y\right|^{2\kappa-\nu}\,dy<\infty$.

The last integral is finite, since its integrand behaves at zero
asymptotically as $C|y|^{2\kappa(k-1)-\nu}$, at the point
$e=\frac x{|x|}\in S_{\nu-1}$ as $C_2|y-e|^{2\kappa-\nu}$ and at
infinity as $C_3|y|^{2\kappa k-2\nu}$. Relations~(6.7) and~(6.8)
imply that
$$
\align
D(n)&\le C'\int |\tilde\chi_0(x)|^2|x|^{2\kappa k-\nu}\,dx
\le C''\int |x|^{2\kappa k-\nu}\prod_{l=1}^\nu\frac1{1+|x^{(l)}|^2}\,dx\\
&\le C'''\int_{|x^{(1)}|=\max\limits_{1\le l\le\nu}|x^{(l}|}
|x^{(1)}|^{2\kappa k-\nu}\prod_{l=1}^\nu\frac1{1+|x^{(l)}|^2}\,dx\\
&=\sum_{p=0}^\infty C'''
\int_{|x^{(1)}|=\max\limits_{1\le l\le\nu}|x^{(l}|,
\;2^p\le |x^{(1)}|<2^{p+1}}
+C'''\int_{|x^{(1)}|=\max\limits_{1\le l\le\nu}|x^{(l}|, |x^{(1)}|<1}.
\endalign
$$
The second term in the last sum can be simply bounded by a constant, since
$B=\left\{x\colon\;|x^{(1)}|=\max\limits_{1\le l\le\nu}|x^{(l}|, \;
|x^{(1)}|<1\right\}\subset\{x\colon\;|x|\le\sqrt \nu\}$,
and $|x^{(1)}|^{2\kappa k-\nu}\prodd_{l=1}^\nu\frac1{1+|x^{(l)}|^2}
\le\const |x|^{2\kappa k-\nu}$ on the set~$B$. Hence
$$
D(n)\le C_1\sum_{p=0}^\infty 2^{p(2\kappa k-\nu)}
\[\int_{-\infty}^\infty\frac 1{1+x^2}\,dx\]^\nu+C_2<\infty.
$$
We have $|\varphi(x)|\le C(1+|x^2|)^{-p}$ with some $C>0$ and $D>0$
if $\varphi\in\Cal S$.
The proof of the estimate $D(\varphi)<\infty$ for $\varphi\in\Cal S$
is similar but simpler.

\medskip\noindent
{\it Proof of part b).}\/ Define, similarly to the function
$J_{\kappa,k}$,
$$
J_{\kappa,k,a}(x)=\int_{x_1+\cdots+x_k=x}
|x_1|^{2\kappa-\nu}a\(\frac{x_1}{|x_1|}\)\cdots|x_k|^{2\kappa-\nu}
a\(\frac{x_k}{|x_k|}\)
\,dx_1\dots\,dx_k, \quad x\in R^{\nu},
$$
where $\,dx_1\dots\,dx_k$ denotes the Lebesgue measure on the
hyperplane $x_1+\cdots+x_k=x$. Since
$$
J_{\kappa,k,a}(x)\ge\int_{y\colon\; |y|<(\frac12+\alpha)|x|,\;
|y-x|<(\frac12+\alpha)|x|} J_{\kappa,k-1,a}(y)
a\(\frac{x-y}{|x-y|}\)|x-y|^{2\kappa-\nu}\,dy
$$
with an arbitrary $\alpha>0$ an argument similar to the one in
part~a) shows that
$$
J_{\kappa,k,a}(x)\left\{
\aligned
&\ge \bar C(\kappa,k)|x|^{2\kappa k-\nu}
\quad\text{if }\frac\nu k>2\kappa>0,\\
&=\infty\qquad \text{if }\kappa\le0 \text{ or } 2\kappa\ge\frac\nu k
\endaligned \right.
$$
if $\frac x{|x|}$ is close to such a point~$x_0\in S_{\nu-1}$
in whose small neighbourhood the function $a(\cdot)$ is separated
from zero. Since $|\tilde\chi_n(x)|^2>0$ for almost all $x\in R^\nu$,
$$
D(n)=\int|\tilde \chi_n(x)|^2J_{\kappa,k,a}(x)\,dx=\infty
$$
under the conditions of part~b). Similarly $D(\varphi)=\infty$
if $|\tilde\varphi(x)|^2>0$ for almost all $x\in R^\nu$. We remark
that the conditions in part~b) can be weakened. It would have been
enough to assume that $a(x)>0$ on a set of positive Lebesgue
measure in~$S_{\nu-1}$.

\medskip
Theorem 6.2 and $6.2'$ together with Proposition 6.3 have the
following

\medskip\noindent
{\bf Corollary 6.4.} {\it The formulae
$$
\align
\xi_n=\sum_{k=1}^M \int\tilde\chi_n(x_1+\cdots+x_k)
\prod_{l=1}^k
\(|x_l|^{-\kappa+(\nu-\alpha)/k}\cdot b_k\(\frac{x_l}{|x_l|}\)\)
&Z_G(\,dx_1)\dots Z_G(\,dx_k),\\
&\qquad n\in\text{\BBB Z}_\nu,
\endalign
$$
and
$$
\align
\xi(\varphi)=\sum_{k=1}^M \int\tilde\varphi(x_1+\cdots+x_k)
\prod_{l=1}^k\(|x_l|^{-\kappa+(\nu-\alpha)/k}\cdot
b_k\(\frac{x_l}{|x_l|}\)\) &Z_G(\,dx_1)\dots Z_G(\,dx_k), \\
&\qquad \varphi\in\Cal S,
\endalign
$$
define self-similar random fields with self-similarity
parameter~$\alpha$ if $G$ is defined by formula~(6.6), the
parameter~$\alpha$ satisfies the inequality $\frac\nu2<\alpha<\nu$,
and the functions $a(\cdot)$ (in the definition of the
measure~$G(\cdot)$ in~(6.6)), $b_1(\cdot)$,\dots $b_k(\cdot)$ are
bounded even functions on $S_{\nu-1}$.}

\medskip
The following observation may be useful when we want to prove
Corollary~6.4. We can replace $\xi_n$ by another random field with
the same distribution. Thus we can write, by exploiting Theorem~4.4,
$$
\xi_n=\sum_{k=1}^M\tilde\chi_n(x_1+\cdots+x_k)
Z_{G'}(\,dx_1)\dots Z_{G'}(\,dx_k),
\quad n\in\text{\BBB Z}_\nu,
$$
with random spectral measure $Z_{G'}$ corresponding to the spectral
measure
$G'(\,dx)=b(\frac x{|x|})^2|x|^{-2\kappa+2(\nu-\alpha)/k}G(\,dx)
=a(\frac x{|x|})b(\frac x{|x|})^2|x|^{-\nu+2(\nu-\alpha)/k}\,dx$.
In the case of generalized random fields a similar argument can be
applied.

\medskip\noindent
{\it Remark 6.5.} The estimate on $J_{\kappa,k}$ and the end of the of
part~a) in Proposition~(6.3) show that the self-similar random fields
$$ \allowdisplaybreaks
\align
\xi(\varphi)&=\sum_{k=1}^M\int \tilde\varphi(x_1+\cdots+x_k)
|x_1+\cdots+x_k|^p \, u\(\frac{x_1+\cdots+x_k}{|x_1+\cdots+x_k|}\)\\
&\qquad \prod_{l=1}^k\(|x_l|^{-\kappa+(\nu-\alpha)/k}\cdot
b_k\(\frac{x_l}{|x_l|}\)\) Z_G(\,dx_1)\dots Z_G(\,dx_k),
\quad \varphi\in\Cal S,
\endalign
$$
and
$$
\align
\xi_n&=\sum_{k=1}^M\int \tilde\chi_n(x_1+\cdots+x_k)
|x_1+\cdots+x_k|^p \, u\(\frac{x_1+\cdots+x_k}{|x_1+\cdots+x_k|}\)\\
&\qquad \prod_{l=1}^k\(|x_l|^{-\kappa+(\nu-\alpha)/k}\cdot
b_k\(\frac{x_l}{|x_l|}\)\) Z_G(\,dx_1)\dots Z_G(\,dx_k),
\quad n\in\text{\BBB Z}_\nu,
\endalign
$$
are well defined if $G$ is defined by formula~(6.6), $a(\cdot)$,
$b(\cdot)$ and $u(\cdot)$ are bounded even functions on $S_{\nu-1}$,
$\frac\nu2<\alpha<\nu$, and $\alpha-p<\nu$ in the generalized and
$\frac{\nu-1}2<\alpha-p<\nu$ is the discrete random field case. The
self-similarity parameter of these random fields is $\alpha-p$. We
remark that in the case $p>0$  this class of self-similar fields
also contains self-similar fields with self-similarity parameter
less than~$\frac\nu2$.

\medskip
In proving the statement of Remark~6.5 we have to check the
integrability conditions needed for the existence of the
Wiener--It\^o integrals $\xi(\varphi)$ and $\xi_n$. To check them
it is worth remarking that in the proof of part~a) of
Proposition~6.2 we proved the estimate $J_{\bar\kappa,k}(x)\le
C(\bar\kappa,k)|x|^{2\bar\kappa k-\nu}$. We want to apply this
inequality in the present case with the choice
$\bar\kappa=\frac{\nu-\alpha}k$. Then arguing similarly to the proof
of part~a) of Proposition~6.2 we get to the problem whether the
relations $\int|\tilde\chi_n(x)|^2|x|^{2p+2(\nu-\alpha)-\nu}\,dx<\infty$
and $\int|\tilde\varphi(x)|^2|x|^{2p+2(\nu-\alpha)-\nu}\,dx<\infty$ if
$\varphi\in\Cal S$ hold under the conditions of Remark~6.5. They
can be proved by the argument applied at the end of
the proof of part~a) of Proposition~6.2.

\medskip
The following question arises in a natural way. When do different
formulas satisfying the conditions of Theorem~6.2 or Theorem~$6.2'$
define self-similar random fields with different distributions?
In particular: Are the self-similar random fields constructed via multiple
Wiener--It\^o integrals necessarily non-Gaussian? We cannot give a
completely satisfactory answer for the above question, but our
former results yield some useful information. Let us substitute the
spectral measure $G$ by $G'$ such that
$\frac{G(\,dx)}{G'(\,dx)}=|g^2(x)|^2$, $g(-x)=\overline{g(x)}$ and the
functions $|x_l|^{-\kappa+(\nu-\alpha)/k}b(\frac{x_l}{|x_l|})$ by
$(\frac{x_l}{|x_l|})g(x_l)|x_l|^{-\kappa+(\nu-\alpha)/k}$ in
Corollary~6.4. By Theorem~4.4 the new field has the same distribution
as the original one. On the other hand, Corollary~5.4 helps us to
decide whether two random variables have different moments, and
therefore different distributions. Let us consider e.g. a moment of
odd order of the random variables $\xi_n$ or $\xi(\varphi)$
defined in Corollary~6.4. It is clear that all $h_\gamma\ge0$.
Moreover, if $b_k(x)$ does not vanish for some even number~$k$, then
there exists a $h_\gamma>0$ in the sum expressing an odd moment of
$\xi_n$ or $\xi(\varphi)$. Hence the odd moments of $\xi_n$ or
$\xi(\varphi)$ are positive in this case. This means in particular
that the self-similar random fields defined in Corollary~6.4 are
non-Gaussian if $b_k$ is non-vanishing for some even~$k$. The next
result shows that the tail behaviour of multiple Wiener--It\^o
integrals of different order is different.

\medskip\noindent
{\bf Theorem 6.6.} {\it Let $G$ be a spectral measure and $Z_G$ a
random spectral measure corresponding to~$G$. For all $h\in\Cal H_G^m$
there exist some constants $K_1>K_2>0$ and $x_0>0$ depending on
the function~$h$ such that
$$
e^{-K_1x^{2/m}}\le P(|I_G(h)|>x)\le e^{-K_2x^{2/m}}
$$
for all $x>x_0$.}

\medskip\noindent
{\it Remark.}\/ As the proof of Theorem~6.6 shows the constant~$K_2$
in the upper bound of the above estimate can be chosen as
$K_m=C_m (EI_G(h)^2)^{-1/m}$ with a constant~$C_m$ depending only on
the order~$m$ of the Wiener--It\^o integral of~$I_G(h)$. This means
that for a fixed number~$m$ the constant~$K_2$ in the above estimate
can be chosen as a constant depending only on the variance of the
random variable~$I_G(h)$. On the other hand, no simple
characterization of the constant~$K_1>0$ appearing in the lower
bound of this estimate is known.

\medskip\noindent
{\it Proof of Theorem 6.6.} {\it a) Proof of the upper estimate.}

We have
$$
P(|I_G(h)|>x)\le x^{2N}E(I_G(h)|^{2N}).
$$
By Corollary~5.6
$$
E(I_G(h)|^{2N})\le \bar C(m,N)[E(I_G(h)^2)]^N\le \bar C(m,N) C_1^N,
$$
and by a simple combinatorial argument we obtain that
$$
\bar C(m,N)\le\frac{(2Nm-1)(2Nm-3)\cdots 1}{(m!\,)^N},
$$
since the numerator on the right-hand side of this inequality
equals the number of complete diagrams
$|\bar\Gamma(\undersetbrace{2N \text{ times }}\to{m,\dots,m})|$
 if vertices from the same row can also be connected. Multiplying the
inequalities
$$
(2nM-2j-1)(2Nm-2j-1-2N)\cdots (2Nm-2j-1-2N(m-1))\le (2N)^mm!,
$$
$j=1,\dots,N$, we obtain that
$$
\bar C(m,N)\le (2N)^{mN}.
$$
(This inequality could be sharpened, but it is sufficient for our
purpose.) Choose a sufficiently small number $\alpha>0$, and define
$N=[\alpha x^{2/m}]$, where $[\cdot]$ denotes integer part. With
this choice we have
$$
P(|I_G(h)|>x)\le(x^{-2}(2\alpha)^mx^2)^N C_1^N=[C_1(2\alpha)^m]^N
\le e^{-K_2x^{2/m}},
$$
if $\alpha$ is chosen in such a way that $C_1(2\alpha)^m\le\frac1e$,
$K_2=\frac\alpha2$, and $x>x_0$ with an appropriate $x_0>0$.

\medskip\noindent
{\it b) Proof of the lower estimate.}

First we reduce this inequality to the following statement. Let
$Q(x_1,\dots,x_k)$ be a homogeneous polynomial of order~$m$ (the
number~$k$ is arbitrary), and $\xi=(\xi_1,\dots,\xi_k)$ a
$k$-dimensional standard normal variable. Then
$$
P(Q(\xi_1,\dots,\xi_k)>x)\ge e^{-Kx^{2/m}} \tag6.9
$$
if $x>x_0$, where the constants $K>0$ and $x_0>0$ may depend on the
polynomial~$Q$.

By the results of Section~4, $I_G(h)$ can be written in the form
$$
I_G(h)=\sum_{j_1+\cdots+j_l=m} C^{k_1,\dots,k_l}_{j_1,\dots,j_l}
H_{j_1}(\xi_{k_1})\cdots H_{j_k}(\xi_{k_l}),
\tag6.10
$$
where $\xi_1,\xi_2,\dots$ are independent standard normal random
variables, $C^{k_1,\dots,k_l}_{j_1,\dots,j_l}$ are
appropriate coefficients, and the right-hand side of~(6.10) is
convergent in~$L_2$ sense. Let us fix a sufficiently large
integer~$k$, and let us consider the conditional distribution of
the right-hand side of~(6.10) under the condition
$\xi_{k+1}=x_{k+1},\xi_{k+2}=x_{k+2},\dots$, where the numbers
$x_{k+1},x_{k+2},\dots$ are arbitrary. This conditional distribution
coincides with the distribution of the random variable
$Q(\xi_1,\dots,\xi_k,x_{k+1},x_{k+2},\dots)$ with probability~1,
where the polynomial $Q$ is obtained by substituting $\xi_{k+1}=x_{k+1}$,
$\xi_{k+2}=x_{k=2},\dots$ into the right-hand side of~(6.10).
It is clear that all these polynomials
$Q(\xi_1,\dots,\xi_k,x_{k+1},x_{k+2},\dots)$ are of order $m$ if
$k$ is sufficiently large. It is sufficient to prove that
$$
P(|Q(\xi_1,\dots,\xi_k,x_{k+1},x_{k+2},\dots)|>x)\ge e^{-Kx^{2/m}}
$$
for $x>x_0$, where the constants $K>0$ and $x_0>0$ may depend on
the polynomial~$Q$. Write
$$
Q(\xi_1,\dots,\xi_k,x_{k+1},x_{k+2},\dots)=
Q_1(\xi_1,\dots,\xi_k)+Q_2(\xi_1,\dots,\xi_k)
$$
where $Q_1$ is a homogeneous polynomial of order~$m$, and~$Q_2$ is a
polynomial of order less than~$m$. The polynomial $Q_2$ can be
rewritten as the sum of finitely many Wiener--It\^o integrals with
multiplicity less than~$m$. Hence the already proved part of
Theorem~6.6 implies that
$$
P(Q_2(\xi_1,\dots,\xi_k)>x)\le e^{-\bar qKx^{2/(m-1)}} .
$$
(We may assume that $m\ge2$). Then an application of relation~(6.9)
to~$Q_1$ implies the remaining part of Theorem~6.6, thus it
suffices to prove~(6.9).

If $Q(\cdot)$ is a polynomial of $k$ variables, then there exist
some $\alpha>0$ and $\beta>0$ such that
$$
\lambda\(\left|Q\(\frac{x_1}{|x|},\dots,\frac{x_k}{|x|}\)\right|>\alpha\)
>\beta,
$$
where $|x|^2=\summ_{j=1}^kx_j^2$, and $\lambda$ denotes the Lebesgue
measure on the $k$-dimensional unit sphere $S_{k-1}$. Exploiting
that $|\xi|$ and $\frac\xi{|\xi|}$ are independent, $\frac\xi{|\xi|}$
is uniformly distributed on the unit sphere $S_{k-1}$, and
$P(|\xi|>x)\ge ce^{-x^2}$ for a $k$-dimensional standard normal
random variable, we obtain that
$$
P(|Q(\xi_1,\dots,\xi_k)|>x)\ge P\(|\xi|^m>\frac x\alpha\)\beta>e^{-Kx^{2/m}},
$$
if the constants $K$ and $x$ are sufficiently large. Theorem~6.6 is
proved.

\medskip
Theorem~6.6 implies in particular that Wiener--It\^o integrals of
different multiplicity have different distributions. A bounded
random variable measurable with respect to the $\sigma$-algebra generated
by a stationary Gaussian field can be expressed as a sum of
multiple Wiener--It\^o integrals. Another consequence of Theorem~6.6
is the fact that the number of terms in this sum must be infinite.

In Theorems~6.2 and~$6.2'$ we have defined a large class of
self-similar fields. The question arises whether this class contains
self-similar fields such that the distributions of their random
variables tend to  one (or zero) at infinity (at minus infinity)
much faster than the normal distribution functions do. This
question has been unsolved by now. By Theorem~6.6 such fields, if
any, must be expressed as a sum of infinitely many Wiener--It\^o
integrals. The above question is of much greater importance
than it may seem at first instant. Some considerations suggest
that in some important models of statistical physics self-similar
fields with very fast decreasing tail distributions appear as
limit, when the so-called renormalization group transformations are
applied for the probability measure describing the state of the
model at critical temperature. (The renormalization group
transformations are the transformations over the distribution of
stationary fields induced by formula~(1.1) or~(1.3),
when $A_N=N^\alpha$, $A(t)=t^\alpha$ with some~$\alpha$.) No
rigorous proof about the existence of such self-similar fields is
known yet. Thus the real problem behind the above question is whether
the self-similar fields interesting for statistical physics can be
constructed via multiple Wiener--It\^o integrals.

\beginsection 7. On the original Wiener--It\^o integral.

In this section the definition of the original Wiener--It\^o
integral introduced by It\^o in~[18] is explained. As the
arguments are very similar to those of Sections~4 and~5 (only the
notations become simpler) most proofs will be omitted.

Let a measure space $(M,\Cal M,\mu)$ with a $\sigma$-finite
measure~$\mu$ be given. Let $\mu$ satisfy the following continuity
property: For all $\e>0$ and $A\in\Cal M$, $\mu(A)<\infty$, there
exist some disjoint sets $B_j\in\Cal M$, $j=1,\dots,N$, with some
integer~$N$ such that $\mu(B_j)<\e$ for all $1\le j\le N$, and
$A=\bigcupp_{j=1}^NB_j$. We introduce the following definition.

\medskip\noindent
{\bf Definition of (Gaussian) random orthogonal measures.} {\it A
system of random variables $Z_\mu(A)$, $A\in\Cal M$,
$\mu(A)<\infty$, is called a Gaussian random orthogonal measure
corresponding to  the measure~$\mu$ if

\medskip
\item{(i)} $Z_\mu(A_1),\dots,Z_\mu(A_k)$ are independent Gaussian
random variables if the sets $A_j\in\Cal M$, $\mu(A_j)<\infty$,
$j=1,\dots,k$, are disjoint.
\item{(ii)} $EZ_\mu(A)=0$, $EZ_\mu(A)^2=\mu(A)$.
\item{(iii)} $Z_\mu\(\bigcupp_{j=1}^k A_j\)=\summ_{j=1}^k Z_\mu(A_k)$
with probability~1 if $A_1,\dots,A_k$ are disjoint sets.

}

\medskip\noindent
{\it Remark.}\/ There is the following equivalent version for the
definition of random orthogonal measures: The system of random variables
system of random variables $Z_\mu(A)$, $A\in\Cal M$, $\mu(A)<\infty$, is
a Gaussian random orthogonal measure corresponding to  the
measure~$\mu$ if

\medskip
\item{(i$'$)} $Z_\mu(A_1),\dots,Z_\mu(A_k)$ are (jointly) Gaussian
random variables for all sets $A_j\in\Cal M$, $\mu(A_j)<\infty$,
$j=1,\dots,k$.
\item{(ii$'$)} $EZ_\mu(A)=0$, and $EZ_\mu(A)Z_\mu(B)=\mu(A\cap B)$
if $A,\,B\in\Cal M$, $\mu(A)<\infty$, $\mu(B)<\infty$.

\medskip
It is not difficult to see that properties~(i),~(ii) and~(iii) imply
relations~(i$'$) and~(ii$'$). On the other hand, it is clear that
(i$'$) and (ii$'$) imply~(i) and~(ii). To see that they also imply
relation~(iii) observe that under these conditions
$$
E\[Z_\mu\(\bigcupp_{j=1}^k A_j\)-\summ_{j=1}^k Z_\mu(A_k)\]^2=0
$$
if $A_1,\dots,A_k$ are disjoint sets.

The second characterization of random orthogonal measures may help
to show that for any measure space $(M,\Cal M,\mu)$ with a
$\sigma$-finite measure~$\mu$ there exists a Gaussian random
orthogonal measure corresponding to  the measure~$\mu$. The main
point in checking this statement is the proof that for any
sets $A_1,\dots,A_k\in\Cal M$, $\mu(A_j)<\infty$, $1\le j\le k$,
there exists a Gaussian random vector $(Z_\mu(A_1),\dots,Z_\mu(A_k))$,
$EZ_\mu(A_j)=0$, with correlation $EZ_\mu(A_i)Z_\mu(A_j)=\mu(A_i\cap A_j)$
for all $1\le i,j\le k$. To prove this we have to show that the
corresponding covariance matrix is really positive definite, i.e.
$\summ_{i,j} c_i\bar c_j\mu(A_i\cap A_j)\ge0$ for an arbitrary vector
$(c_1,\dots,c_k)$. But this follows from the observation
$\summ_{i,j} c_i\bar c_j\chi_{A_i\cap A_j}(x)
=\summ_{i,j} c_i\bar c_j\chi_{A_i}(x)\overline{\chi_{ A_j}(x)}
=\left|\summ_i c_i\chi_{A_i}(x)\right|^2\ge0$ for all $x\in M$, if we
integrate this inequality with respect to the measure~$\mu$ in the
space~$M$.

\medskip
We define the real Hilbert spaces $\bar{\Cal  K}^n_\mu$, $n=1,2,\dots$.
The space $\bar{\Cal K}^n_\mu$ consists of the real-valued measurable
functions over
$(\undersetbrace{n\text{ times}}\to{M\times\cdots\times M},\,
\undersetbrace{n\text{ times}}\to{\Cal M\times\cdots\times\Cal M})$
such that
$$
\|f\|^2=\int|f(x_1,\dots,x_n)|^2\mu(\,dx_1)\dots\mu(\,dx_n)<\infty,
$$
and the last formula defines the norm in $\bar{\Cal K}^n_\mu$. Let
$\Cal K^n_\mu$ denote the subspace of $\bar{\Cal K}^n_\mu$ consisting
of the functions $f\in\bar{\Cal K}^n_\mu$ such that
$$
f(x_1,\dots,x_n)=f(x_{\pi(1)},\dots,x_{\pi(n)})
\quad\text{for all }\pi\in\Pi_n.
$$
Let the spaces $\bar{\Cal K}^0_\mu$ and $\Cal K^0_\mu$ consist of the
real constants with the norm $\|c\|=|c|$. Finally we define the
Fock space $\text{Exp}\,\Cal K_\mu$ which consists of the sequences
$f=(f_0,f_1,\dots)$, $f_n\in\Cal K^n_\mu$, $n=0,1,2,\dots$, such that
$$
\|f\|^2=\sum_{n=0}^\infty \frac1{n!} \|f_n\|^2<\infty.
$$

Given a random orthogonal measure $Z_\mu$ corresponding to $\mu$, let
us introduce the $\sigma$-algebra
$\Cal F=\sigma(Z_\mu(A)\colon\; A\in\Cal M,\,\mu(A)<\infty)$. Let
$\Cal K$ denote the real Hilbert space of square integrable random
variables measurable with respect to the $\sigma$-algebra~$\Cal F$.
Let $\Cal K_{\le n}$ denote the subspace that is the closure of the
linear space containing the polynomials of the random variables
$Z_\mu(A)$ of order less than or equal to~$n$. Let $\Cal K_n$ be the
orthogonal completion of $\Cal K_{\le n-1}$ to $\Cal K_{\le n}$.
(The norm is defined as $\|\xi\|^2=E\xi^2$ in these Hilbert spaces.)

The multiple Wiener--It\^o integrals with respect to the random
orthogonal measure $Z_\mu$, to be defined below, give a unitary
transformation from $\text{Exp}\,\Cal K_\mu$ to $\Cal K$. We shall
denote these integrals by $\int'$ to distinguish them from the
Wiener--It\^o integrals defined in Section~4.

First we define the class of elementary functions
$\hat{\bar{\Cal K}}_\mu^n\subset\bar{\Cal K}_\mu^n$. A function
$f\in\bar{\Cal K}_\mu^n$ is in $\hat{\bar{\Cal K}}_\mu^n$ if there
exists a finite system of disjoint sets $\Delta_1,\dots,\Delta_N$,
with $\Delta_j\in\Cal M$, $\mu(\Delta_j)<\infty$, \ $j=1,\dots,N$,
such that $f(x_1,\dots,x_n)$ is constant on the sets
$\Delta_{j_1}\times\cdots\times\Delta_{j_n}$ if the indices
$j_1,\dots,j_n$ are disjoint, and $f(x_1,\dots,x_n)$ equals zero
outside these sets. We define
$$
\int' f(x_1,\dots,x_n)Z_\mu(\,dx_1)\dots Z_\mu(\,dx_n)
=\sum f(x_{j_1},\dots,x_{j_n})Z_\mu(\Delta_{j_1})\cdots Z_\mu(\Delta_{j_n})
$$
for $f\in\hat{\bar{\Cal K}}_\mu^n$, where $x_k\in\Delta_k$, $k=1,\dots,N$.

Let $\hat{\Cal K}_\mu^n=\hat{\bar{\Cal K}}_\mu^n\cap \Cal K_\mu^n$. The
random variables
$$
I'_\mu(f)=\frac1{n!}\int'f(x_1,\dots,x_n)Z_\mu(\,dx_1)\dots Z_\mu(\,dx_n),
\quad f\in\hat{\bar{\Cal K}}_\mu^n,
$$
have zero expectation, integrals of different order are orthogonal,
$$
I'_\mu(f)=I'_\mu(\,\text{Sym}\, f), \quad\text{and }
\text{Sym}\,f\in \hat{\Cal K}_\mu^n \text{ if }
f\in\hat{\bar{\Cal K}}_\mu^n,
$$
$$
EI'_\mu(f)^2\le \frac1{n!}\|f\|^2 \quad \text{if }
f\in\hat{\bar{\Cal K}}_\mu^n, \tag7.1
$$
and~(7.1) holds with equality if $f\in\hat{\Cal K}_\mu^n$.

It can be seen that $\hat{\bar{\Cal K}}^n_\mu$ is dense
in~$\bar{\Cal K}_\mu^n$, hence relation~(7.1) enables us to extend the
definition of the $n$-fold Wiener--It\^o integrals
over~$\bar{\Cal K}_\mu^n$. All the above mentioned relations remain
valid if $f\in\hat{\bar{\Cal K}}_\mu^n$ is substituted by
$f\in\bar{\Cal K}_\mu^n$, and $f\in\hat{\Cal K}_\mu^n$ is substituted
by $f\in\Cal K_\mu^n$. We formulate It\^o's formula for these integrals.
It can be proved similarly to Theorem~4.2.

\medskip\noindent
{\bf Theorem~7.1. (It\^o's formula)} {\it Let
$\varphi_1,\dots,\varphi_m$, $\varphi_j\in\Cal K^1_\mu$ for all
$1\le j\le m$, be an orthonormal system in $L^2_\mu$. Let some
positive integers $j_1,\dots,j_m$ be given, put $j_1+\cdots+j_m=N$,
and define for all $i=1,\dots,N$
$$
g_i=\varphi_1 \text{ for } 1\le i\le j_1,\quad\text{and \ }
g_i=\varphi_s \quad\text{for } j_1+\cdots+j_{s-1}<i\le j_1+\cdots+j_s.
$$
Then
$$
\align
&H_{j_1}\(\int'\varphi_1(x)Z_\mu(\,dx)\)\cdots
H_{j_m}\(\int'\varphi_m(x)Z_\mu(\,dx)\)\\
&\qquad=\int' g_1(x_1)\cdots g_N(x_N)\,Z_\mu(\,dx_1)\dots Z_\mu(\,dx_N)\\
&\qquad=\int'\text{\rm Sym}\,
[ g_1(x_1)\cdots g_N(x_N)]\,Z_\mu(\,dx_1)\dots Z_\mu(\,dx_N).
\endalign
$$
}

\medskip
(Let me remark that the diagram formula (Theorem~5.3) also remains
valid for this integral if we replace $-x_j$ is by $x_j$ and
$G(\,dx_j)$ by $\mu(\,dx_j)$, $N-2|\gamma|+1\le j\le N-|\gamma|$,
in the definition of~$h_\gamma$ in formula~(5.1).)

\medskip
It can be seen with the help of ~Theorem~7.1 that the transformation
$I'_\mu\colon\;\text{Exp}\,\Cal K_\mu\to\Cal K$, where
$I'_\mu(f)=\summ_{n=0}^\infty I'_\mu(f_n)$, \
$f=(f_0,f_1,\dots)\in\,\text{Exp}\,{\Cal K}_\mu$ is a unitary
transformation, and so are the transformations $(n!)^{1/2}I'_\mu$
from $\Cal K_\mu^n$ to~$\Cal K_n$.

\medskip
Let us consider the special case
$(M,\Cal M,\mu)=(R^\nu,\Cal B^\nu,\lambda)$, where $\lambda$ denotes
the Lebesgue measure in~$R^\nu$. A random orthogonal measure
corresponding to $\lambda$ is called the white noise. A random
{\it spectral measure}\/ corresponding to $\lambda$, when the
Lebesgue measure is considered as the spectral measure of a
generalized field, is also called a white noise. The next result,
that can be considered as a random Plancherel formula, establishes
a connection between the two types of Wiener--It\^o integrals with
respect to white noise.

\medskip\noindent
{\bf Proposition 7.2.} {\it Let
$f=(f_0,f_1,\dots,)\in\,\text{\rm Exp}\,\Cal K_\lambda$ be an element
of the Fock space corresponding to the Lebesgue measure in the
Euclidean space $(R^\nu,\Cal B^\nu)$. Then
$f'=(f'_0,f'_1,\dots,)\in\,\text{\rm Exp}\,\Cal H_\lambda$ with the
functions $f'_0=f_0$ and $f'_n=(2\pi)^{-n\nu/2}\tilde f_n$,
$n=1,2,\dots$, (where
$\tilde f_n(u_1,\dots,u_n)=
\int_{R^{n\nu}}e^{i(x,u)}f_n(x_1,\dots,x_n)\,dx_1\dots\,dx_n$ with
$x=(x_1,\dots,x_n)$ and $u=(u_1,\dots,u_n)$),
and
$$
\align
&\sum_{n=0}^\infty \frac1{n!}\int' f_n(x_1,\dots,x_n)\,
Z_\lambda(\,dx_1)\dots Z_\lambda(\,dx_n) \\
&\qquad \overset\Delta\to=\sum_{n=0}^\infty \frac1{n!}
\int f'_n(u_1,\dots,u_n)\, Z_\lambda(\,du_1)\dots Z_\lambda(\,du_n),
\endalign
$$
where $Z_\lambda(\,dx)$ is a white noise as a random orthogonal
measure and $Z_\lambda(\,du)$ is a white noise as a random spectral
measure.}

\medskip\noindent
{\it Proof of Proposition 7.2.}\/ We have
$$
(2\pi)^{-n\nu/2}\|\tilde f_n\|_{L_\lambda^2}=\|f_n\|_{L_\lambda^2},
$$
hence $f'\in\,\text{Exp}\,\Cal H_\lambda$.

Let $\varphi_1,\varphi_2,\dots$ be a complete orthonormal system
in~$L_\lambda^2$. Then $\varphi'_1,\varphi'_2,\dots$ is also a
complete orthonormal system in $L_\lambda^2$, and if
$$
f_n(x_1,\dots,x_n)
=\sum c_{j_1,\dots,j_n}\varphi_{j_1}(x_1)\cdots\varphi_{j_n}(x_n),
$$
then
$$
f'_n(u_1,\dots,u_n)
=\sum c_{j_1,\dots,j_n}\varphi'_{j_1}(u_1)\cdots\varphi'_{j_n}(u_n).
$$
Hence an application of It\^o's formula for both types of integrals,
(i.e. Theorems~4.2 and~7.1) imply Proposition~7.2.

\medskip\noindent
Finally we restrict ourselves to the case $\nu=1$. We formulate a
result which reflects a connection between multiple Wiener--It\^o
integrals and classical It\^o integrals. Let $W(t)$,
$a\le t\le b$, be a Wiener process, and let us define the random
orthogonal measure $Z(\,dx)$ as
$$
Z(A)=\int\chi_A(x)W(\,dx), \quad A\subset[a,b),\quad A\in\Cal B^1.
$$
Then we have the following

\medskip\noindent
{\bf Proposition 7.3.} {\it Let $f\in\Cal K^n_{\lambda[a,b)}$, where
$\lambda[a,b)$ denotes the Lebesgue measure on the interval~$[a,b)$.
Then
$$
\align
&\int' f(x_1,\dots,x_n)\,Z(\,dx_1)\dots Z(\,dx_n) \tag7.2 \\
&\qquad=n!\int_a^b \(\int_a^{t_n}\(\cdots \(\int_a^{t_3}\(\int_a^{t_2}
f(t_1,\dots,t_n)W(\,dt_1)\)W(\,dt_2)\)\dots\) W(\,dt_n)\).
\endalign
$$
}

\medskip\noindent
{\it Proof of Proposition 7.3.}\/ Given a function
$f\in\hat{\Cal K}^n_{\lambda[a,b)}$, let the function $\hat f$ be
defined as
$$
\hat f(x_1,\dots,x_n)=\left\{
\aligned
&f(x_1,\dots,x_n)\quad\text{if \ }x_1<x_2<\cdots<x_n \\
&0 \quad \text{otherwise.}
\endaligned \right.
$$
It is not difficult to check Proposition~7.3 for such special
functions $f\in\hat{\Cal K}^n_{\lambda[a,b)}$ for which the
function $\hat f$is the indicator function of a rectangle of
the form $\prodd_{j=1}^n[a_j,b_j)$ with constants
$a\le a_1<b_1<a_2<b_2<\cdots<a_n<b_n\le b$. Here we exploit the
relation $I'(f)=n!I'(\hat f)$. Beside this, we have to calculate
the value of the right-hand side of formula~(7.2) for such
elementary functions~$f\in\hat{\Cal K}^n_{\lambda[a,b)}$. A
simple inductive argument shows that it equals
$\prodd_{j=1}^n[W(b_j)-W(a_j)]$ if
$a\le a_1<b_1<a_2<b_2<\cdots<a_n<b_n\le b$, and it equals zero
otherwise. Then a simple limiting procedure with the help
of the approximation of general functions in
$\Cal K^n_{\lambda[a,b)}$ by the linear combinations of
such functions proves Proposition~7.3 in the general case.

\medskip
As a consequence of Proposition~7.3 in the case $\nu=1$ multiple
Wiener--It\^o integrals can be substituted by It\^o integrals in the
investigation of most problems. In the case $\nu=2$ there is no simple
definition of It\^o integrals. On the other hand, no problem arises in
generalizing the definition of multiple Wiener--It\^o integrals to the
case $\nu\ge2$.

\beginsection 8. Non-central limit theorems.

In this section we investigate the problem formulated in Section~1, and
we show how the technique of Wiener--It\^o integrals can be applied
for the investigation of such a problem. We restrict ourselves to the
case of discrete fields, although the case of generalized fields can
be discussed in almost the same way. The proof of some details will be
omitted. They can be found in~[9]. First we recall the following

\medskip\noindent
{\bf Definition 8A. (Definition of slowly varying functions.)} {\it
A function $L(t)$, $t\in[t_0,\infty)$, $t_0>0$, is said to be a
slowly varying function (at infinity) if
$$
\lim_{t\to\infty}\frac{L(st)}{L(t)}=1 \quad\text{for all \ } s>0.
$$
}

\medskip
We shall apply the following description of slowly varying functions.

\medskip\noindent
{\bf Theorem 8A. (Karamata's theorem.)} {\it If a slowly varying function
$L(t)$ is bounded on every finite interval, then it can be represented
in the form
$$
L(t)=a(t)\exp\left\{\int_{t_0}^t \frac{\e(s)}s\,ds\right\},
$$
where $a(t)\to a_0\neq0$, and $\e(t)\to0$ as $t\to\infty$, and the
functions $a(\cdot)$ and $\e(\cdot)$ are bounded in every finite
interval.}

\medskip
Let $X_n$, $n\in\,\text{\BBB Z}_\nu$, be a stationary Gaussian field
with expectation zero and  a correlation function
$$
r(n)=EX_0X_n=|n|^{-\alpha}a\(\frac n{|n|}\)L(|n|),
\quad n\in\text{\BBB Z}_\nu, \tag8.1
$$
where $0<\alpha<\nu$, $L(t)$ is a slowly varying function, bounded
in all finite intervals, and $a(t)$ is a continuous function on
the unit sphere $\Cal S_{\nu-1}$, satisfying the symmetry property
$a(x)=a(-x)$ for all $x\in\Cal S_{\nu-1}$. Let $G$ denote the
spectral measure of the field~$X_n$, and let us define the
measures~$G_N$, $N=1,2,\dots$, by the formula
$$
G_N(A)=\frac{N^\alpha}{L(N)}G\(\frac AN\),\quad A\in\Cal B^\nu,
\quad N=1,2,\dots. \tag8.2
$$

Now we recall the definition of vague convergence of not necessarily
finite measures on a Euclidean space.

\medskip\noindent
{\bf Definition of vague convergence of measures.} {\it Let $G_n$,
$n=1,2,\dots$, be a sequence of locally finite measures over $R^\nu$,
i.e. let $G_n(A)<\infty$ for all measurable bounded sets~$A$. We say
that the sequence $G_n$ vaguely converges to a locally finite
measure~$G_0$ (in notation $G_n\overset{v}\to\rightarrow G_0$) if
$$
\lim_{n\to\infty}\int f(x)\,G_n(\,dx)=\int f(x)\,G_0(\,dx)
$$
for all continuous functions~$f$ with a bounded support.}

\medskip
We formulate the following

\medskip\noindent
{\bf Lemma 8.1.} {\it Let $G$ be the spectral measure of a stationary
field with a correlation function $r(n)$ of the form~(8.1). Then the
sequence of measures~$G_N$ defined in~(8.2) tends vaguely to a
locally finite measure~$G_0$. The measure~$G_0$ has the homogeneity
property
$$
G_0(A)=t^{-\alpha}G_0(tA) \quad \text{for all } A\in\Cal B^\nu
\quad\text{and } t>0, \tag8.3
$$
and it satisfies the identity
$$
\aligned
&2^\nu\int e^{i(t,x)}\prod_{j=1}^\nu\frac{1-\cos x^{(j)}}{(x^{(j)})^2}
\,G_0(\,dx) \\
&\qquad =\int_{[-1,1]^\nu} (1-|x^{(1)}|)\cdots (1-|x^{(\nu)}|)
\frac{a\(\frac{x+t}{|x+t|}\)}{|x+t|^\alpha}\,dx, \quad\text{for all }
t\in R^\nu.
\endaligned \tag8.4
$$
}

\medskip
We postpone the proof of Lemma~8.1 for a while.

Formulae~(8.3) and~(8.4) imply that the function~$a(t)$ and the
number~$\alpha$ in the definition~(8.1) of a correlation
function~$r(n)$ uniquely determine the measure~$G_0$. Indeed,
by formula~(8.4) they determine the (finite) measure
$\prodd_{j=1}^\nu\frac{1-\cos x^{(j)}}{(x^{(j)})^2}G_0(\,dx)$, since
they determine its Fourier transform. Hence they also determine the
measure~$G_0$. (Formula~(8.3) shows that this is a locally finite
measure). Let us also remark that since $G_N(A)=G_N(-A)$ for all
$N=1,2,\dots$ and $A\in \Cal B^\nu$, the relation $G_0(A)=G_0(-A)$,
$A\in\Cal B^\nu$ also holds. These properties of the measure~$G_0$
imply that it can be considered as the spectral measure of a
generalized random field. Now we formulate

\medskip\noindent
{\bf Theorem 8.2.} {\it Let $X_n$, $n\in\text{\BBB Z}_\nu$, be a
stationary Gaussian field with a correlation function $r(n)$
satisfying relation~(8.1). Let us define the stationary random
field $\xi_j=H_k(X_j)$, $j\in\text{\BBB Z}_\nu$, with some positive
integer~$k$, where $H_k(x)$ denotes the $k$-th Hermite polynomial
with leading coefficient~1, and assume that the parameter~$\alpha$
appearing in~(8.1) satisfies the relation $0<\alpha<\frac\nu k$.
If the random fields $Z^N_n$, $N=1,2,\dots$, $n\in\text{\BBB Z}_\nu$,
are defined by formula~(1.1) with $A_N=N^{\nu-k\alpha/2}L(N)^{k/2}$
and the above defined $\xi_j=H_k(X_j)$, then their multi-dimensional
distributions tend to those of the random field~$Z^*_n$,
$$
Z^*_n=\int \tilde\chi_n(x_1+\cdots+x_k)\,
Z_{G_0}(\,dx_1)\dots Z_{G_0}(\,dx_k), \quad n\in\text{\BBB Z}_\nu.
$$
Here $Z_{G_0}$ is a random spectral measure corresponding to the
spectral measure $G_0$ which appeared in Lemma~8.1. The function
$\tilde\chi_n(\cdot)$, $n=(n^{(1)},\dots,n^{(\nu)})$, is (similarly
to Section~6) the Fourier transform of the uniform distribution on
the $\nu$-dimensional unit cube
$\prod\limits_{p=1}^\nu[n^{(p)},n^{(p)}+1]$.}

\medskip\noindent
{\it Remark.}\/ The condition that the correlation function~$r(n)$
of the random field $X_n$, $n\in\text{\BBB Z}_\nu$, satisfies
formula~(8.1) can be weakened. Theorem~8.2 and Lemma~8.1 remain
valid if~(8.1) is replaced by the slightly weaker condition
$$
\lim_{T\to\infty}\sup_{n\colon\;n\in\text{\BBB Z}_\nu,\,|n|\ge T}
\frac{r(n)}{|n|^{-\alpha}a\(\frac n{|n|}\)L(|n|)}=1,
$$
where $0<\alpha<\nu$, $L(t)$ is a slowly varying function, bounded
in all finite intervals, and $a(t)$ is a continuous function on the
unit sphere $\Cal S_{\nu-1}$, satisfying the symmetry property
$a(x)=a(-x)$ for all $x\in\Cal S_{\nu-1}$.

\medskip
First we explain why the choice of the normalizing constant~$A_N$ in
Theorem~8.2 was natural, then we explain the ideas of the proof,
finally we work out the details.

Corollary~5.5 implies in particular  that
$EH_k(\xi)H_k(\eta)=k! (E\xi\eta)^k$ for a Gaussian random vector
$(\xi,\eta)$ with $E\xi=E\eta=0$ and $E\xi^2=E\eta^2=1$. Hence
$$
E(Z_n^N)^2=\frac{k!}{A_N^2}\sum_{j\in B_0^N,\,l\in B_0^N}r(j-l)^k
\sim\frac{k!}{A_N^2}
\sum_{j,\,l\in B_0^N}|j-l|^{-k\alpha}a^k\(\frac{j-l}{|j-l|}\)L(|j-l|)^k,
$$
with the set $B_0^N$ introduced after formula~(1.1). Some calculation
with the help of the above formula shows that with our choice of~$A_N$
the expectation $E(Z_n^N)^2$ is separated both from zero and infinity,
therefore this is the natural norming factor. In this calculation we
have to exploit the condition $k\alpha<\nu$, which implies that in the
sum expressing $E(Z_n^N)^2$ those terms are dominant for which $j-l$
is relatively large, more explicitly it is of order~$N$. There are
$\const N^{2\nu}$ such terms.

The field $\xi_n$ is subordinated to the Gaussian field~$X_n$. It is
natural to rewrite it in canonical form, and to express $Z_n^N$ via
multiple Wiener--It\^o integrals. It\^o's formula yields the relation
$$
\xi_n=H_k\(\int e^{i(n,x)}Z_G(\,dx)\)=\int e^{i(n,x_1+\cdots+x_k)}
Z_G(\,dx_1)\dots Z_G(\,dx_k),
$$
where $Z_G$ is the random spectral measure adapted to the random
field~$X_n$. Then
$$
\align
Z_n^N&=\frac1{A_N}\sum_{j\in B_n^N}\int e^{i(j,x_1+\cdots+x_k)}
Z_G(\,dx_1)\dots Z_G(\,dx_k)\\
&=\frac1{A_N}\int e^{i(Nn,x_1+\cdots+x_k)}\prod_{j=1}^\nu
\frac{e^{iN(x_1^{(j)}+\cdots+x_k^{(j)})}-1}
{e^{i(x_1^{(j)}+\cdots+x_k^{(j)})}-1}\,
Z_G(\,dx_1)\dots Z_G(dx_k).
\endalign
$$
Let us make the substitution $y_j=Nx_j$, $j=1,\dots,k$, in the last
formula, and let us rewrite it in a form resembling formula~($6.5'$).
To this end, let us introduce the measures~$G_N$ defined in~(8.2).
By Lemma~4.5 we can write
$$
Z_n^N\overset\Delta\to=\int f_N(y_1,\dots,y_k)\tilde\chi_n(y_1+\cdots+y_k)\,
Z_{G_N}(\,dy_1)\dots Z_{G_N}(dy_k)
$$
with
$$
f_N(y_1,\dots,y_k)=\prod_{j=1}^\nu\frac{i(y_1^{(j)}+\cdots+y_k^{(j)})}
{\(\exp\left\{i\frac1N(y_1^{(j)}+\cdots+y_k^{(j)})\right\}-1\)N}.
\tag8.5
$$
(It follows from Lemma~8B formulated below and the Fubini theorem that
the set, where the denominator of the function $f_N$ disappears,
i.e. the set where $y_1^{(j)}+\cdots+y_k^{(j)}=2lN\pi$ with some integer
$l\neq0$ and $1\le j\le\nu$ has 0 $G_N\times\cdots\times G_N$ measure.
This means that the functions $f_N$ are well defined.)
The functions~$f_N$ tend to 1 uniformly in all bounded regions, and the
measures~$G_N$ tend vaguely to~$G_0$ as $N\to\infty$ by Lemma~8.1. These
relations suggest the following limiting procedure. The limit of $Z_n^N$
can be obtained by substituting $f_N$ with~1 and $G_N$ with~$G_0$ in
the Wiener--It\^o integral expressing~$Z_n^N$. We want to justify
this formal limiting procedure. For this we have to show that the
Wiener--It\^o integral expressing~$Z_n^N$ is essentially concentrated
in a large bounded region independent of ~$N$. The $L_2$~isomorphism
of Wiener--It\^o integrals can help us in showing that. The next
lemma is a useful tool for the justification of  the above limiting
procedure.

Before formulating this lemma we make a small digression. It was
explained that Wiener--It\^o integrals can be defined also with
respect to random stationary fields $Z_G$ adapted to a stationary
Gaussian random field whose spectral measure~$G$ may have atoms, and
we can work with them similarly as in the case of non-atomic
spectral measures. Here a lemma will be proved which shows that in
the proof of Theorem~8.2 we do not need this observation, because
if the correlation function of the random field satisfies~(8.1), then
its spectral measure is non-atomic.

\medskip\noindent
{\bf Lemma~8B.} {\it Let the correlation function of a stationary
field $X_n$, $n\in\text{\BBB Z}_\nu$, satisfy the relation
$r(n)\le A|n|^{-\alpha}$ with some $A>0$ and $\alpha>0$ for all
$n\in\text{\BBB Z}_\nu$, $n\neq0$. Then its spectral measure $G$ is
non-atomic. Moreover, all hyperplanes $\summ_{j=1}^\nu c_jx^{(j)}=d$
defined with some constants $c_j$ and $d$ have zero $G$ measure.}

\medskip\noindent
{\it Proof of Lemma 8B.} Lemma 8B clearly holds if $\alpha>\nu$,
because in this case the spectral measure~$G$ has even a density
 function $g(x)=\summ_{n\in\text{\BBB Z}_\nu}e^{-i(n,x)}r(n)$.
On the other hand, the $p$-fold convolution of the spectral measure
$G$ with itself (on the torus $R^\nu/2\pi\text{\BBB Z}_\nu$) has
Fourier transform, $r(n)^p$, hence in the case $p>\frac\nu\alpha$
it is non-atomic. Hence it is enough to show that if the
convolution $G*G$ is a non-atomic measure, then so is the
measure~$G$. But this is obvious, because if there were a point
$x\in R^\nu/2\pi\text{\BBB Z}_\nu$ such that $G(\{x\})>0$, then
$G*G(\{x+x\})>0$ would hold, and this is a contradiction.
(Here addition is taken on the torus.) The proof of the zero $G$
measure of all hyperplanes is similar.

\medskip
Now we formulate the following result.

\medskip\noindent
{\bf Lemma~8.3.} {\it Let $G_N$, $N=1,2,\dots$, be a sequence
of spectral measures on $R^\nu$ tending vaguely to a spectral
measure~$G_0$. Let a sequence of measurable functions
$K_N=K_N(x_1,\dots,x_k)$, $N=0,1,2,\dots$, be given such that
$K_N\in\bar{\Cal H}_{G_N}^k$ for $N=1,2,\dots$. Assume further
that the following properties hold: For all $\e>0$ there exist
some constants $A=A(\e)>0$ and $N_0=N_0(\e)>0$ and finitely many
rectangles $P_1,\dots,P_M$ with some cardinality $M=M(\e)$ on
$R^{k\nu}$ which satisfy the following conditions~a) and~b). (We
call a set $P\in\Cal B^{k\nu}$ a rectangle if it can be written
in the form $P=L_1\times\cdots\times L_k$ with some bounded open
sets $L_s\in\Cal B^\nu$, $1\le s\le k$, with boundaries
$\partial L_s$ of zero $G_0$~measure, i.e.\
$G_0(\partial L_s)=0$ for all $1\le s\le k$.)

\medskip
\item{a)} The function $K_0$ is continuous on the set
$B=[-A,A]^{k\nu}\setminus\bigcupp_{j=1}^MP_j$, and $K_N\to K_0$
uniformly on the set $B$ as $N\to\infty$. Beside this the hyperplanes
$x_p=\pm A$ have zero $G_0$~measure for all $1\le p\le\nu$.
\item{b)} $\int_{R^{k\nu}\setminus B}|K_N(x_1,\dots,x_k)|^2
G_N(\,dx_1)\dots G_N(dx_k)<\frac{\e^3}{k!}$ if $N=0$ or $N\ge N_0$,
and $K_0(-x_1,\dots,-x_k)=\overline{K_0(x_1,\dots,x_k)}$ for all
$(x_1,\dots,x_k)\in R^{k\nu}$.

\medskip
Then $K_0\in\bar{\Cal H}_{G_0}^k$, and
$$
\int K_N(x_1,\dots,x_k)\,Z_{G_N}(\,dx_1)\dots Z_{G_N}(\,dx_k)
\overset{\Cal D}\to\rightarrow
\int K_0(x_1,\dots,x_k)\,Z_{G_0}(\,dx_1)\dots Z_{G_0}(\,dx_k)
$$
as $N\to\infty$, where $\overset{\Cal D}\to\rightarrow$ denotes
convergence in distribution.}

\medskip\noindent
{\it Remark.}\/ In the proof of Theorem~8.2 or of its generalization
Theorem~$8.2'$ formulated later a simpler version of Lemma~8.3 with a
simpler proof would suffice. We could work with such a version where
the rectangles~$P_j$ do not appear. We formulated this somewhat more
complicated result, because it can be applied in the proof of more
general theorems, where the limit is given by such a Wiener--It\^o
integral whose kernel function may have discontinuities.

\medskip\noindent
{\it Proof of Lemma~8.3.}\/ Conditions~a) and~b) obviously imply that
$$
\int|K_0(x_1,\dots,x_k)|^2\,G_0(\,dx_1)\dots G_0(\,dx_k)<\infty,
$$
hence $K_0\in\bar{\Cal H}_{G_0}^k$. By using the same argument as
in the definition of Wiener--It\^o integrals with atomic spectral
measure we can reduce the lemma to the case when the spectral
measures $G_N$, $N=0,1,2,\dots$, are non-atomic.

Let us fix an $\e>0$, and let $A>0$, $N_0>0$ and the rectangles
$P_1,\dots,P_M$ satisfy conditions~a) and~b) with this~$\e$. Then
$$
\aligned
&E\left[\int [1-\chi_B(x_1,\dots,x_k)]K_N(x_1,\dots,x_k)\,
Z_{G_N}(\,dx_1)\dots Z_{G_N}(\,dx_k)\right]^2\\
&\qquad \le k!\int_{R^{k\nu}\setminus B}|K_N(x_1,\dots,x_k)|^2
G_N(\,dx_1)\dots G_N(\,dx_k)<\e^3
\endaligned \tag8.6
$$
for $N=0$ or $N>N_0$, where $\chi_B$ denotes the indicator function
of the set~$B$ introduced in the formulation of condition~a).

Since $B\subset [-A,A]^{k\nu}$, and $G_N\overset v\to\rightarrow G_0$,
hence $G_N\times\cdots\times G_N(B)<C(A)$ with an appropriate constant
$C(A)<\infty$ for all $N=0,1,\dots$.  Because of this estimate and
the uniform convergence $K_N\to K_0$ on the set~$B$ we have
$$
\aligned
&E\left[\int(K_N(x_1,\dots,x_k)-K_0(x_1,\dots,x_k))\chi_B(x_1,\dots,x_k)\,
Z_{G_N}(\,dx_1)\dots Z_{G_N}(\,dx_k)\right]^2\\
&\qquad\le k!\int_B
|K_N(x_1,\dots,x_k)-K_0(x_1,\dots,x_k)|^2\,
G_N(\,dx_1)\dots G_N(\,dx_k)<\e^3
\endaligned \tag8.7
$$
for $N>N_1$ with some $N_1=N_1(A,\e)$.

With the help of formulas~(8.6) and~(8.7) we reduce the proof of
Lemma~8.3 to that of the relation
$$
\aligned
&\int K_0(x_1,\dots,x_k)\chi_B(x_1,\dots,x_k)\,
Z_{G_N}(\,dx_1)\dots Z_{G_N}(\,dx_k)\\
&\qquad \overset{\Cal D}\to\rightarrow
\int K_0(x_1,\dots,x_k)\chi_B(x_1,\dots,x_k)\,
Z_{G_0}(\,dx_1)\dots Z_{G_0}(\,dx_k).
\endaligned  \tag8.8
$$

We do this with the help of a classical result of probability
theory about the basic properties of the so-called Prokhorov
metric defined in the following way. Given a complete separable
metric space $(X,\Cal A)$ with some metric~$\rho$ let $\Cal S$
denote the space of probability measures on it. The Prokhorov
metric $\rho_P$ is the metric in the space $\Cal S$ defined by
the formula
$\rho_P(\mu,\nu)=\inf\{\e\colon\; \mu(A)\le\nu(A^\e)+\e\text{ for all }
A\in\Cal A\}$ for two probability measures $\mu,\nu\in\Cal S$, where
$A^\e=\{x\colon\;\rho(x,A)<\e\}$. It is known that $\rho_P$ is a
metric on~$\Cal S$ (in particular $\rho_P(\mu,\nu)=\rho_P(\nu,\mu)$)
which metricizes the weak convergence of probability measures in
the metric space~$(X,\Cal A)$. (see R.M.~Dudley Distances of
probability measures and random variables. Ann.~Math.~Statist.~39,
1563--1572 (1968)).

I formulated the above result for probability measures in a general
metric space, but I shall work on the real line. Given a random
variable~$\xi$ let $\mu(\xi)$ denote its distribution.Put
$\xi_N=k!I_{G_N}(K_N(x_1,\dots,x_k))$, $N=0,1,2,\dots$. With such
a notation we can formulate the statement of Lemma~8.3 in the
following way. For all $\e>0$ there exists some index
$N'_0=N'_0(\e)$ such that $\rho_P(\mu(\xi_N),\mu(\xi_0))\le4\e$
for all~$N\ge N'_0$.

To prove this statement let us first show that for three random
variables $\xi$, $\bar\xi$ and~$\eta$ such that
$P(|\eta|\ge\e)\le\e$ the inequality
$\rho_P(\mu(\xi+\eta),\mu(\bar\xi))\le\rho_P(\mu(\xi),\mu(\bar\xi))+\e$
holds. Indeed, since $\{\oo\colon\;\xi(\oo)+\eta(\oo)\in A\}\subset
\{\oo\colon\;\xi(\oo)\in A^\e\}\cup\{\oo\colon\;|\eta(\oo)|\ge\e\}$,
we have $P(\xi+\eta\in A)\le P(\xi\in A^\e)+\e$ for any
set~$A\in\Cal B_1$ if $P(|\eta|\ge\e)\le\e$. Beside this,
$P(\xi\in A^\e)\le P(\bar\xi\in A^{\e+\delta})+\delta$ for all
$\delta>\rho_P(\mu(\xi),\mu(\bar\xi))$. Hence
$P(\xi+\eta\in A)\le P(\bar\xi\in A^{\e+\delta})+\e+\delta$ for all
$A\in\Cal B_1$ and $\delta>\rho_P(\mu(\xi),\mu(\bar\xi))$, i.e.\
$\rho_P(\mu(\xi+\eta),\mu(\bar\xi))\le\e+\delta$, and this implies
the inequality
$\rho_P(\mu(\xi+\eta),\mu(\bar\xi))\le\rho_P(\mu(\xi),\mu(\bar\xi))+\e$.

Put
$$
\align
\xi_N^{(1)}&=k!I_{G_N}(K_0(x_1,\dots,x_k)\chi_B(x_1,\dots,x_k)),\\
\xi_N^{(2)}&=k!I_{G_N}(K_N(x_1,\dots,x_k)-K_0(x_1,\dots,x_k))
\chi_B(x_1,\dots,x_k)),\\
\xi_N^{(3)}&=k!I_{G_N}(1-\chi_B(x_1,\dots,x_k))K_N(x_1,\dots,x_k))
\endalign
$$
for all $N=0,1,2,\dots$. With this notation it follows from
relation~(8.8) and the
fact that the Prokhorov metric metricizes the weak convergence
that $\rho_P(\mu(\xi_N^{(1)}),\mu(\xi_0^{(1)}))\le\e$ if $N\ge N'_1(\e)$
with some threshold index~$N'_1(\e)$. Formulas~(8.6) and~(8.7)
together with the Chebishev inequality imply that
$P(|\xi_N^{(2)}|\ge\e)\le\e$ and $P(|\xi_N^{(3)}|\ge\e)\le\e$
if $N\ge N'_2(\e)$ or $N=0$ with some threshold index~$N'_2(\e)$.
Beside this, we have $\xi_0=\xi_0^{(1)}+\xi_0^{(3)}$ and
$\xi_N=\xi_N^{(1)}+\xi_N^{(2)}+\xi_N^{(3)}$ for $N=1,2,\dots$.
The above mentioned properties of the random variables we considered
together with the result of the previous paragraph imply that
$$
\align
\rho_P(\mu(\xi_N),\mu(\xi_0))
&=\rho_P(\mu(\xi_N^{(1)}+\xi_N^{(2)}+\xi_N^{(3)}),
\mu(\xi_0^{(1)}+\xi_0^{(3)}))\\
&\le\rho_P(\mu(\xi_N^{(1)}+\xi_N^{(2)}+\xi_N^{(3)}),
\mu(\xi_0^{(3)}))+\e \\
&\le\rho_P(\mu(\xi_N^{(2)}+\xi_N^{(3)}),\mu(\xi_0^{(3)}))+2\e\\
&\le\rho_P(\mu(\xi_N^{(3)}),\mu(\xi_0^{(3)}))+3\e\le 4\e
\endalign
$$
if $N\ge N'_0(\e)=\max(N'_1(\e),N'_2(\e))$, and this is what we
wanted to prove.

To prove~(8.8) we will show that
$K_0(x_1,\dots,x_k)\chi_B(x_1,\dots,x_k)$ can be well approximated by
functions from $\hat{\bar{\Cal H}}_{G_0}^k$ in the following sense.
For all $\e>0$ there exists such an (elementary) function
$f_\e\in\hat{\bar{\Cal H}}^k_{G_0}$ for which the $L^2_{G_0^k}$
norm of the difference
$f_\e(x_1,\dots,x_k)-K_0(x_1,\dots,x_k)\chi_B(x_1,\dots,x_k)$ is less
than~$\frac{\e^3}{k!}$, and also the $L_{G_N^k}^2$ norm of this
difference is smaller than~$\frac{\e^3}{k!}$ if $N\ge N_2$ with some
threshold~$N_2=N_2(\e)$. Moreover, the function $f_\e$ has the
following additional property. The function
$f_\e\in\hat{\bar{\Cal H}}_{G_0}^k$ is adapted to such a regular
system $\Cal D=\{\Delta_j,\;j=\pm1,\dots,\pm M\}$ for which the
boundaries of the sets $\Delta_j$ satisfy the relation
$G_0(\partial\Delta_j)=0$ for all $j=\pm1,\dots,\pm M$.

First I claim that such a function $f_\e\in\hat{\bar{\Cal H}}_{G_0}^k$
satisfies the relation
$$
\int f_\e(x_1,\dots,x_k)\,Z_{G_N}(\,dx_1)\dots Z_{G_N}(\,dx_k)
\overset{\Cal D}\to\rightarrow
\int f_\e(x_1,\dots,x_k)\,Z_{G_0}(\,dx_1)\dots Z_{G_0}(\,dx_k)\tag8.9
$$
as $N\to\infty$. To prove~(8.9) observe that for the
regular system $\Cal D=\{\Delta_j,\;j=\pm1,\dots,\pm M\}$
to which the function $f_\e\in\hat{\bar{\Cal H}}_{G_0}^k$ is
adapted has the following property: The (Gaussian)
random vectors $(Z_{G_N}(\Delta_j),\;j=\pm1,\dots,\pm M)$
converge in distribution to the (Gaussian) random vector
$(Z_{G_0}(\Delta_j),\;j=\pm1,\dots,\pm M)$ as $N\to\infty$.
(We needed at this point the property $G_0(\partial\Delta_j)=0$.
It follows from the vague convergence
$G_N\overset v \to\rightarrow G_0$, (similarly to the case of weak
convergence) that $\limm_{N\to\infty}G_N(\Delta_j)=G_0(\Delta_j)$ if
$G_0(\partial\Delta_j)=0$, which also implies the weak convergence
of the above mentioned random vectors, but the condition
$G_0(\partial\Delta_j)=0$ cannot be dropped here.) Beside this,
the Wiener--It\^o integrals in formula~(8.9) are polynomials (not
depending on the parameter~$N$) of these random vectors. Hence
relation~(8.9) holds.

The approximation result formulated after formula~(8.8) implies
the existence of such a function $V_\e\in\bar{\Cal H}_G^k$
for all $\e>0$ for which
$$
k!I_G(K_0(x_1,\dots,x_k)\chi_B(x_1,\dots,x_k))
=k!I_G(f_\e(x_1,\dots,x_k))+k!I_G(V_\e(x_1,\dots,x_k))
$$
with the above considered function~$f_\e$, and
$$
\align
&E\(\int V_\e(x_1,\dots,x_k)Z_{G_N}(\,dx_1)\dots Z_{G_N}(\,dx_k)\)^2\\
&\quad \le k!\int(K_0(x_1,\dots,x_k)\chi_B(x_1,\dots,x_k)-
f_\e(x_1,\dots,x_k))^2G_N(\,dx_1)\dots G_N(\,dx_k)\le\e^3
\endalign
$$
if $N\ge N_2$ with some $N_2=N_2(\e)$ or $N=0$. The last inequality
together with formula~(8.9) imply~(8.8) in the same way as
inequalities~(8.6), (8.7) and~(8.8) imply Lemma~8.3.

We still have to prove that $K_0(x_1,\dots,x_k)\chi_B(x_1,\dots,x_k)$
can be well approximated by an appropriate elementary
function~$f_\e$. This can be reduced with help of the relation
$G_N\overset v\to\rightarrow G_0$ to the following simpler statement
where only the limit measure~$G_0$ is considered. For all $\e>0$
there is a function $f_\e\in\hat{\bar{\Cal H}}_{G_0}^k$ for which
$\int|K_0(x_1,\dots,x_k)\chi_B(x_1,\dots,x_k)-f_\e(x_1,\dots,x_k)|^2
G_0(\,dx_1)\dots G_0(\,dx_k)<\frac{\e^3}{k!}$, and it is adapted to
such a regular system $\Cal D=\{\Delta_j,\;j=\pm1,\dots,\pm \bar N\}$
whose elements $\Delta_j$ have boundaries of zero $G_0$ measure.

Indeed, as
$|K_0(x_1,\dots,x_k)\chi_B(x_1,\dots,x_k)-f_\e(x_1,\dots,x_k)|^2$
is a bounded function with a compact support which is continuous
in almost all points with respect to the measure
$\undersetbrace k\to{G_0\times\dots\times G_0}$ the relation
$G_N\overset v\to\rightarrow G_0$ together with the above statement
about~$G_0$ also implies the inequality
$$
\int|K_0(x_1,\dots,x_k)\chi_B(x_1,\dots,x_k)-f_\e(x_1,\dots,x_k)|^2
G_N(\,dx_1)\dots G_N(\,dx_k)<\frac{\e^3}{k!}
$$
if $N\ge N_2(\e)$.

The existence of the approximating function
$f_\e\in\hat{\bar{\Cal H}}_{G_0}^k$ with the desired properties
can be shown similarly to proof of the result that
$\hat{\bar{\Cal H}}_G^n$ is a dense subset of $\bar{\Cal H}_G^n$.
We can reduce the statement we want to verify first to a slightly
modified version of Statement~A and then to a slightly modified
version of Statement~B in the proof of the result about the
good approximability of a function in~$\bar{\Cal H}_G^n$ by a
function in~$\hat{\bar{\Cal H}}_G^n$. In the modified version of
Statement~A we are dealing with such sets~$A$ and~$A_1$ whose
boundaries have zero~$G_0^k$ measures, and we demand the same
property about the set $B\in\Cal B^{k\nu}$ appearing in their
approximation. Similarly, in the modified version of Statement~B
we are dealing with such sets $D_j$ whose boundaries have zero
$G_0$~measure, and we demand that the set $F\in\Cal B^{k\nu}$
providing a good approximation also must have boundaries of zero
$G_0^k$ measure. This can be proved similarly to the original
statements. We leave to the reader to work out the details.

\medskip\noindent
{\it Remark.} We have formulated this statement in the case when $G_N$ is
a spectral measure on $R^\nu$. But it remains valid if $G_N$ is a
spectral measure on the torus of size $2C_N\pi$ with $C_N\to\infty$
if $N\to\infty$ if we identify this torus with the set
$[-C_N\pi,C_N\pi)^\nu\subset R^\nu$ in a natural way.

\medskip
Now we turn to the proof of Theorem~8.2.

\medskip\noindent
{\it The proof of Theorem~8.2.} We want to prove that for all
positive integers $p$, real numbers $c_1,\dots,c_p$  and
$n_l\in\text{\BBB Z}_\nu$, $l=1,\dots,p$,
$$
\sum_{l=1}^p c_l Z^N_{n_l}\overset{\Cal D}\to\rightarrow
\sum_{l=1}^p c_lZ^*_{n_l},
$$
since this relation also implies the convergence of the
multi-dimensional distributions. Applying the same calculation
as before we get with the help of Lemma~4.5 that
$$
\sum_{l=1}^p c_lZ_{n_l}^N=\frac1{A_N}\sum_{l=1}^p c_l\int
\sum_{j\in B_{n_l}^N}e^{i(j,x_1+\cdots+x_k)}\,Z_G(\,dx_1)\dots Z_G(\,dx_k),
$$
and
$$
\sum_{l=1}^p c_lZ^N_{n_l}\overset\Delta\to=\int K_N(x_1,\dots,x_k)\,
Z_{G_N}(\,dx_1)\dots Z_{G_N}(\,dx_k)
$$
with
$$
\aligned
K_N(x_1,\dots,x_k)&=\frac1{N^\nu}\sum_{l=1}^p c_l\sum_{j\in B_{n_l}^N}
\exp\left\{i\(\frac jN,x_1+\cdots+x_k\)\right\}\\
&=f_N(x_1,\dots,x_k)\sum_{l=1}^p c_l\tilde\chi_{n_l}(x_1+\cdots+x_k).
\endaligned \tag8.10
$$
with the function~$f_N$ defined in~(8.5) and the measure~$G_N$ defined
in~(8.2).

Let us define the function
$$
K_0(x_1,\dots,x_k)=\sum_{l=1}^p c_l\tilde\chi_{n_l}(x_1+\cdots+x_k)
$$
and the measures~$\mu_N$ on $R^{k\nu}$ by the formula
$$
\mu_N(A)=\int_A|K_N(x_1,\dots,x_k)|^2 G_N(\,dx_1)\dots G_N(\,dx_k),
\quad A\in\Cal B^{k\nu} \text{ and } N=0,1,\dots,
\tag8.11
$$
where $G_0$ is the vague limit of the measures~$G_N$.

To prove Theorem~8.2 it is enough to show that Lemma~8.3 can be applied
with these spectral measures~$G_N$ and functions~$K_N$. (We choose
no exceptional rectangles~$P_j$ in this application of Lemma~8.3.)
Since $G_N\overset v\to\rightarrow G_0$, and $K_N\to K_0$ uniformly
in all bounded regions in $R^{k\nu}$, it is enough to show, beside
the proof of Lemma~8.1, that the measures $\mu_N$, $N=1,2,\dots$,
tend weakly to the (necessary finite) measure~$\mu_0$, (in notation
$\mu_N\overset w\to\rightarrow \mu_0$), i.e.
$\int f(x)\mu_N(\,dx)\to\int f(x)\mu_0(\,dx)$ for all continuous
and bounded functions~$f$ on~$R^{k\nu}$. Then this convergence implies
condition~b) in Lemma~8.3. Moreover, it is enough to show the slightly
weaker statement by which there exists some finite measure
$\bar\mu_0$ such that $\mu_N\overset w\to\rightarrow\bar\mu_0$, since
then $\bar\mu_0$ must coincide with $\mu_0$ because of the relations
$G_N\overset v\to\rightarrow G_0$ and $K_N\to K_0$ uniformly in
all bounded regions of $R^{k\nu}$, and $K_0$ is a continuous function.

There is a well-known theorem in probability theory about the
equivalence between weak convergence of finite measures and the
convergence of their Fourier transforms. It would be natural to
apply this theorem for proving
$\mu_N\overset w\to\rightarrow\bar\mu_0$. On the other hand, we have
the additional information that the measures $\mu_N$, $N=1,2,\dots$,
are concentrated in the cubes $[-N\pi,N\pi)^{k\nu}$, since the
spectral measure~$G$ is concentrated in $[-\pi,\pi)^\nu$. It
is more fruitful to apply a version of the above mentioned theorem,
where we can exploit our additional information. We formulate the
following

\medskip\noindent
{\bf Lemma 8.4.} {\it Let $\mu_1,\mu_2,\dots$ be a sequence of finite
measures on $R^l$ such that $\mu_N(R^l\setminus [-C_N\pi,C_N\pi)^l)=0$
for all $N=1,2,\dots$, with some sequence $C_N\to\infty$ as
$N\to\infty$. Define the modified Fourier transform
$$
\varphi_N(t)=\int_{R^l} \exp\left\{i\(\frac{[tC_N]}{C_N},x\)\right\}
\mu_N(\,dx), \quad t\in R^l,
$$
where $[tC_N]$ is the integer part of the vector $tC_N\in R^l$. (For an
$x\in R^l$ its integer part $[x]$ is the vector $n\in\text{\BBB Z}_l$
for which $x^{(p)}-1<n^{(p)}\le x^{(p)}$ if $x^{(p)}\ge0$, and
$x^{(p)}-1\le n^{(p)}< x^{(p)}+1$ if $x^{(p)}<0$ for all
$p=1,2,\dots,l$.) If for all $t\in R^l$ the sequence $\varphi_N(t)$
tends to a function $\varphi(t)$ continuous at the origin, then the
measures $\mu_N$ weakly tend to a finite measure~$\mu_0$, and
$\varphi(t)$ is the Fourier transform of~$\mu_0$.}

\medskip
I make some comments on the conditions of Lemma~8.4. Let us observe
that if the measures~$\mu_N$ or a part of them are shifted with a
vector $2\pi C_N u$ with some $u\in\text{\BBB Z}_l$, then their
modified Fourier transforms $\varphi_N(t)$ do not change because of
the periodicity of the trigonometrical functions $e^{i(j/C_N,x)}$,
$j\in\text{\BBB Z}_l$. On the other hand, these new measures
which are not concentrated in $[-C_N\pi,C_N\pi)^l$, have no limit.
Lemma~8.4 states that if the measures~$\mu_N$ are concentrated in the
cubes $[-C_N\pi,C_N\pi)^l$, then the convergence of their modified
Fourier transforms defined in Lemma~8.4, which is a weaker condition,
than the convergence of their Fourier transforms, also implies their
convergence to a limit measure.

\medskip\noindent
{\it Proof of Lemma~8.4.}\/ The proof is a natural modification of
the proof about the equivalence of weak convergence of measures and
the convergence of their Fourier transforms. First we show that
for all $\e>0$ there exits some $K=K(\e)$ such that
$$
\mu_N(x\colon\; x\in R^l,\; |x^{(1)}|>K)<\e \quad \text{for all \ }N\ge1.
\tag8.12
$$

As $\varphi(t)$ is continuous at the origin there is some $\delta>0$
such that
$$
|\varphi(0,\dots,0)-\varphi(t,0,\dots,0)|<\frac\e2 \quad\text{if \ }
|t|<\delta. \tag8.13
$$
We have
$$
0\le \Re[\varphi_N(0,\dots,0)-\varphi_N(t,0,\dots,0)]
\le2\varphi_N(0,\dots,0) \tag8.14
$$
for all $N=1,2,\dots$. The sequence in the middle term of~(8.14) tends
to $\Re[\varphi(0,\dots,0)-\varphi(t,0,\dots,0)]$ as $N\to\infty$.
The right-hand side of~(8.14) is a bounded function in the variable~$N$,
since it is convergent. Hence the dominated convergence theorem can
be applied. We get because of the condition $C_N\to\infty$ and
relation~(8.13) that
$$
\align
&\lim_{N\to\infty} \int_0^{[\delta C_N]/C_N} \frac1\delta\,
\Re[\varphi_N(0,\dots,0)-\varphi_N(t,0,\dots,0)]\,dt\\
&\qquad=\int_0^\delta\frac1\delta\,
\Re[\varphi(0,\dots,0)-\varphi(t,0,\dots,0)]\,dt<\frac\e2.
\endalign
$$
Hence
$$
\align
\frac\e2&> \lim_{N\to\infty} \int_0^{[\delta C_N]/C_N}\frac1\delta\,
\Re[\varphi_N(0,\dots,0)-\varphi_N(t,0,\dots,0)]\,dt \\
&=\lim_{N\to\infty}\int
\(\frac1\delta\int_0^{[\delta C_N]/C_N}
\Re [1-e^{i[tC_N]x^{(1)}/C_N}]\,dt\) \mu_N(\,dx)\\
&=\lim_{N\to\infty}\int
\frac1{\delta C_N}\sum_{j=0}^{[\delta C_N]-1}
\Re\[1-e^{ijx^{(1)}/C_N}\]\mu_N(\,dx)\\
&\ge\limsup_{N\to\infty} \int_{\{|x^{(1)}|>K\}}\frac1{\delta C_N}
\sum_{j=0}^{[\delta C_N]-1} \Re \[1-e^{ijx^{(1)}/C_N}\]\mu_N(\,dx)\\
&=\limsup_{N\to\infty}\int_{\{|x^{(1)}|>K\}}\(1-\frac1{\delta C_N}
\Re\frac{1-e^{i[\delta C_N]x^{(1)}/C_N}}
{1-e^{ix^{(1)}/C_N}}\)\mu_N(\,dx)
\endalign
$$
with arbitrary $K>0$. (In the last but one step of this calculation we
have exploited that $\frac1{\delta C_N}\sum\limits_{j=0}^{[\delta C_N]-1}
\Re[1-e^{ijx^{(1)}/C_N}]\ge0$ for all $x^{(1)}\in R^1$.)

Since the measure $\mu_N$ is concentrated in
$\{x\colon\, x\in R^l,\;|x^{(1)}|\le C_N\pi\}$, and
$$
\align
\Re\frac{1-e^{i[\delta C_N]x^{(1)}/C_N}}
{1-e^{ix^{(1)}/C_N}}
&=\frac{\Re\(i e^{-ix^{(1)}/2C_N}
\(1-e^{i[\delta C_N]x^{(1)}/C_N}\)\)}
{i(e^{-ix^{(1)}/2CN}-e^{ix^{(1)}/2C_N})}\\
&\le\frac1{\left|\sin \(\dfrac{x^{(1)}}{2C_N}\)\right|}
\le \frac{C_N\pi}{|x^{(1)}|}
\endalign
$$
if $|x^{(1)}|\le C_N\pi$, (here we exploit that
$|\sin u|\ge\frac2\pi|u|$ if $|u|\le\frac\pi2$), hence we have with
the choice $K=\frac{2\pi}{\delta}$
$$
\frac\e2>\limsup_{N\to\infty}\int_{\{|x^{(1)}|>K\}}
\(1-\left|\frac\pi{\delta x^{(1)}}\right|\)\mu_N(\,dx)
\ge\limsup_{N\to\infty}\frac12\mu_N(|x^{(1)}|>K).
$$
As the measures $\mu_N$ are finite the inequality
$\mu_N(|x^{(1)}|>K)<\e$ holds for each index~$N$ with a
constant~$K=K(N)$ that may depend on~$N$. Hence the above
inequality implies that formula~(8.12) holds for all $N\ge1$
with a possibly larger index~$K$ that does not depend on~$N$.

Applying the same argument to the other coordinates we find that
for all $\e>0$ there exists some $C(\e)<\infty$ such that
$$
\mu_N\(R^l\setminus[-C(\e),C(\e)]^l\)<\e \quad \text{for all } N=1,2,\dots.
$$

Consider the usual Fourier transforms
$$
\tilde\varphi_N(t)=\int_{R^l}e^{i(t,x)}\mu_N(\,dx), \quad t\in R^l.
$$
Then
$$
|\varphi_N(t)-\tilde\varphi_N(t)|\le 2\e+\int_{[-C(\e),C(\e)]}
\left|e^{i(t,x)}-e^{i([tC_N]/C_N,x)}\right|\mu_N(\,dx)\le 2\e
+\frac{lC(\e)}{C_N}\mu_N(R^l)
$$
for all $\e>0$. Hence $\tilde\varphi_N(t)-\varphi_N(t)\to0$ as
$N\to\infty$, and $\tilde\varphi_N(t)\to\varphi(t)$. (Observe that
$\mu_N(R^l)=\varphi_N(0)\to\varphi(0)<\infty$ as $N\to\infty$, hence
the measures~$\mu_N(R^l)$ are uniformly bounded, and $C_N\to\infty$
by the conditions of Lemma~8.4.) Then Lemma~8.4 follows from standard
theorems on Fourier transforms.

\medskip
We return to the proof of Theorem~8.2. We apply Lemma~8.4 with $C_N=N$
and $l=k\nu$ for the measures $\mu_N$ defined in~(8.11). Because of the
middle term in~(8.10) we can write
$$
\varphi_N(t_1,\dots,t_k)=\sum_{r=1}^p\sum_{s=1}^p c_rc_s
\psi_N(t_1+n_r-n_s,\dots,t_k+n_r-n_s)
$$
with
$$
\align
&\psi_N(t_1,\dots,t_r)=\frac1{N^{2\nu}}\int\exp\left\{
i\frac1N((j_1,x_1)+\cdots+(j_k,x_k))\right\}\\
&\qquad\qquad\sum_{p\in B^N_0}\sum_{q\in B^N_0}
\exp\left\{i\(\frac{p-q}N,x_1+\cdots+x_k\)\right\}
G_N(\,dx_1)\dots G_N(\,dx_k)\\
&\qquad=\frac1{N^{2\nu-k\alpha}L(N)^k}\sum_{p\in B_0^N}\sum_{q\in B_0^N}
r(p-q+j_1)\cdots r(p-q+j_k), \tag8.15
\endalign
$$
where  $j_p=[t_pN]$, $t_p\in R^\nu$, $p=1,\dots,k$.

The asymptotical behaviour of $\psi_N(t_1,\dots,t_k)$ for $N\to\infty$
can be investigated by the help of the last relation and formula~(8.1).
Rewriting the last double sum in the form of a single sum by fixing
first the variable $l=p-q\in [-N,N]^\nu\cap\text{\BBB Z}_\nu$, and
then summing up for~$l$ one gets
$$
\psi_N(t_1,\dots,t_k)=\int_{[-1,1]^\nu} f_N(t_1,\dots,t_k,x)\,dx
$$
with
$$
f_N(t_1,\dots,t_k,x)=\(1-\frac{[|x^{(1)}N|]}N\)\cdots
\(1-\frac{[|x^{(\nu)}N|]}N\)
\frac{r([xN]+j_1)}{N^{-\alpha}L(N)}\cdots
\frac{r([xN]+j_k)}{N^{-\alpha}L(N)}.
$$
(In the above calculation we exploited that in the last sum of
formula~(8.15) the number of pairs $(p,q)$ for which
$p-q=l=(l_1,\dots,l_\nu)$ equals $(N-|l_1|)\cdots(N-|l_\nu|)$.)

It can be seen with the help of formula~(8.1) that
$$
f_N(t_1,\dots,t_k,x)\to f_0(t_1,\dots,t_k,x) \tag8.16
$$
with
$$
f_0(t_1,\dots,t_k,x)=(1-|x^{(1)}|)\dots(1-|x^{(\nu)}|)
\frac{a\left(\frac{x+t_1}{|x+t_1|}\right)}{|x+t_1|^\alpha}\dots
\frac{a\left(\frac{x+t_k}{|x+t_k|}\right)}{|x+t_k|^\alpha}
$$
uniformly on the set $x\in[1,1]^\nu\setminus
\bigcupp_{p=1}^k\{x\colon\;|x+t_p|>\e\}$ for all $\e>0$.

We claim that
$$
\psi_N(t_1,\dots,t_k)\to\psi_0(t_1,\dots,t_k)
=\int_{[-1,1]^\nu}f_0(t_1,\dots,t_k,x)\,dx,
$$
and $\psi_0$ is a continuous function.

This relation implies that $\mu_N\overset w\to\rightarrow\mu_0$. To
prove it, it is enough to show beside formula~(8.16) that
$$
\left|\int_{|x+t_p|<\e} f_0(t_1,\dots,t_k,x)\,dx\right|<C(\e),
\quad p=1,\dots,k, \tag8.17
$$
and
$$
\int_{|x+t_p|<\e} |f_N(t_1,\dots,t_k,x)|\,dx<C(\e),
\quad p=1,\dots,k, \text{ and } N=1,2,\dots \tag$8.17'$
$$
with a constant $C(\e)$ such that $C(\e)\to0$ as $\e\to0$.

By H\"older's inequality
$$
\align
\left|\int_{|x+t_p|<\e} f_0(t_1,\dots,t_k,x)\,dx\right|
&\le C\prod_{1\le l\le k,\,l\neq p}\[\int_{x\in[-1,1]^\nu}
|x+t_l|^{-k\alpha}\,dx\]^{1/k} \\
&\qquad \[\int_{|x+t_p|\le\e}|x+t_p|^{-k\alpha}\,dx\]^{1/k}
\le C'e^{\nu/k-\alpha}
\endalign
$$
with some appropriate $C>0$ and $C'>0$, since $\nu-k\alpha>0$,
and $a(\cdot)$ is a bounded function. Similarly,
$$
\align
\int_{|x+t_p|<\e} |f_N(t_1,\dots,t_k,x)|\,dx
&\le \prod_{1\le l\le k,\,l\neq p}\[\int_{x\in[-1,1]^\nu}
\frac{|r([xN]+j_l)|^k}{N^{-k\alpha}L(N)^k}\,dx\]^{1/k},\\
&\qquad \[\int_{|x+t_p|\le\e}
\frac{|r([xN]+j_p)|^k}{N^{-k\alpha}L(N)^k}\,dx\]^{1/k}
\endalign
$$

It is not difficult to see, by using Karamata's theorem, that if
$L(\cdot)$ is a slowly varying function which is bounded in all
finite intervals, then for all $\eta>0$ there is a threshold
index~$N_0$ and a number $C=C(N_0,\eta)$ such that
$$
L(tN)\le Ct^{-\eta}L(N) \quad\text{for all } t<1
\text{ \ and } N\ge N_0.
$$
Hence formula~(8.1) implies that
$$
|r([xN]+j_l)|=|r([xN]+[t_lN])
\le CN^{-\alpha}L(N)(1+|x+t_l|^{-\alpha-\eta}), \tag8.18
$$
and
$$
\align
\int_{|x+t_p|<\e}\frac{|r([xN]+j_p)|^k}{N^{-k\alpha}L(N)^k}\,dx
&\le B\int_{|x+t_p|<\e}(1+|x+t_p|^{-k(\alpha+\eta)})\,dx
\le B'\e^{\nu-k(\alpha +\eta)}\\
\int_{x\in[-1,1]^\nu}\frac{|r([xN]+j_l)|^k}{N^{-k\alpha}L(N)^k}\,dx
&\le B''.
\endalign
$$
for a sufficiently small constant $\eta>0$ with some constants
$B,B',B''<\infty$ depending on $\eta$ and $t_p$, $1\le p\le k$.
(Let us remark that~(8.18) holds also for
$|[xN]+j_l|\le K_1$ with some $K_1>0$ independent of~$N$, i.e. when
the argument of $r(\cdot)$ is relatively small, because
$|r(n)|\le1$ for all $n\in\text{\BBB Z}_\nu$.) Therefore we get, by
choosing an $\eta>0$ such that $k(\alpha+\eta)<\nu$, the inequality
$$
\int_{|x+t_p|<\e} |f_N(t_1,\dots,t_k,x)|\,dx
\le C\e^{\nu/k-(\alpha +\eta)}
$$
with some $C<\infty$. The right-hand side of this inequality tends
to zero as $\e\to0$. Hence we proved beside~(8.16) formulae~(8.17)
and~$(8.17')$, therefore also the relation
$\mu_N\overset w\to\rightarrow \mu_0$. To complete the proof of
Theorem~8.2 it remains to prove Lemma~8.1.

\medskip\noindent
{\it Proof of Lemma~8.1.}\/ Introduce the notation
$$
K_N(x)=\prod_{j=1}^\nu\frac {e^{ix^{(j)}}-1}{N(e^{ix^{(j)}/N}-1)},
\quad N=1,2,\dots,
$$
and
$$
K_0(x)=\prod_{j=1}^\nu\frac {e^{ix^{(j)}}-1}{ix^{(j)}}.
$$
Let us consider the measures $\mu_N$ defined in formula~(8.11) in the
special case $k=1$, $p=1$, $c_1=1$. Then
$$
\mu_N(A)=\int_A |K_N(x)|^2\,G_N(\,dx).
$$
We have already seen in the proof of Theorem~8.2 that
$\mu_N\overset w\to\rightarrow \mu_0$ with some finite measure~$\mu_0$,
and the Fourier transform of~$\mu_0$ is
$$
\varphi_0(t)=\int_{[-1,1]^\nu}(1-|x^{(1)}|)\cdots (1-|x^{(\nu)}|)
\frac{a\(\frac{x+t}{|x+t|}\)}{|x+t|^\alpha}\,dx.
$$
First we show that for all $T\ge1$ there is a finite measure $G_0^T$
concentrated on $[-T\pi,T\pi]^\nu$ such that
$$
\lim_{N\to\infty}\int f(x)\,G_N(\,dx)=\int f(x)\,G_0^T(\,dx) \tag8.19
$$
for all continuous functions~$f$ which vanish outside the cube
$[-T\pi,T\pi]^\nu$.

Let a continuous function~$f$ vanish outside the cube
$[-T\pi,T\pi]^\nu$ with some $T\ge1$. Let $M=[\frac N{2T}]$. Then
$$
\align
\int f(x)G_N(\,dx)&=\frac{N^\alpha}{L(N)}\cdot \frac{L(M)}{M^\alpha}
\int f\(\frac NMx\)G_M(\,dx)\\
&=\frac{N^\alpha L(M)}{M^\alpha L(N)}\int f\(\frac NMx\)
|K_M(x)|^{-2}\mu_M(\,dx)\\
&\qquad\qquad \to (2T)^\alpha\int f(2Tx)|K_0(x)|^{-2}\mu_0(\,dx)\\
&\qquad\qquad\qquad
=\int f(x)\frac{(2T)^\alpha}{|K_0(\frac x{2T})|^2}\mu_0\(\,\frac{dx}{2T}\)
\quad \text{as }N\to\infty,
\endalign
$$
because $f(\frac NMx)|K_M(x)|^{-2}$ vanishes outside the cube
$[-\pi,\pi]^\nu$, $f(\frac NMx)|K_M(x)|^{-2}\to f(2Tx)|K_0(x)^{-2}$
uniformly, (the function~$K_0(\cdot)^{-2}$ is continuous in the cube
$[-\pi,\pi]^\nu$,) and $\mu_M\overset w\to\rightarrow\mu_0$ as
$N\to\infty$. Hence relation~(8.19) holds. The measures $G_0^T$
appearing in~(8.19) are consistent for different parameters~$T$,
i.e.\ $G_0^T$ is the restriction of the measure $G_0^{T'}$ to the
cube $[-T\pi,T\pi]^\nu$ if $T'>T$. It can be seen with the help of
these facts that there is a locally finite measure $G_0$ on $R^\nu$
such that $G_0^T$ is its restriction to the cube $[-T\pi,T\pi]^\nu$,
and $G_N\overset v\to\rightarrow G_0$.

As $G_N\overset v\to\rightarrow G_0$, and $|K_N(x)|^2\to|K_0(x)|^2$
uniformly in all bounded regions, hence
$\mu_N\overset v\to\rightarrow\bar\mu_0$, where
$\bar\mu_0(A)=\int_A|K_0(x)|^2G_0(\,dx)$, $A\in\Cal B^\nu$. Since
$\mu_N\overset w\to\rightarrow\mu_0$ the measures~$\mu_0$
and~$\bar\mu_0$ must coincide, i.e.
$$
\mu_0(A)=\int_A |K_0(x)|^2\,G_0(\,dx), \quad A\in\Cal B^\nu.
$$
Relation~(8.4) expresses the fact that $\varphi_0$ is the Fourier
transform of~$\mu_0$.

Let us extend the definition of the measures~$G_N$ given in~(8.2) to
all non-negative real numbers~$u$. It is easy to see that the
relation $G_u\overset v\to\rightarrow G_0$ as $u\to\infty$ remains
valid. Hence we get for all fixed $s>0$ and continuous functions~$f$
with compact support that
$$
\align
\int f(x) G_0(\,dx)&=\lim_{u\to\infty}\int f(x)\,G_u(\,dx)
=\lim_{u\to\infty}\frac{s^\alpha L(\frac us)}{L(u)}
\int f(sx)G_{\frac us}(\,dx)\\
&=s^\alpha \int f(sx)G_0(\,dx)=\int f(x)s^\alpha G_0\(\frac{dx}s\).
\endalign
$$
This identity implies the homogeneity property~(8.3) of~$G_0$.
Lemma~8.3 is proved.

\medskip
The next result is a generalization of Theorem~8.2.

\medskip\noindent
{\bf Theorem~8.2$'$.} {\it Let $X_n$, $n\in\text{\BBB Z}_\nu$, be a
stationary Gaussian field with a correlation function $r(n)$ defined
in~(8.1). Let $H(x)$ be a real function with the properties
$EH(X_n)=0$ and $EH(X_n)^2<\infty$. Let us consider the Fourier
expansion
$$
H(x)=\sum_{j=1}^\infty c_jH_j(x), \quad \sum c_j^2j!<\infty, \tag8.20
$$
of the function~$H(\cdot)$ by the Hermite polynomials~$H_j$ (with leading
coefficients~1). Let $k$ be the smallest index in this expansion such
that $c_k\neq0$. If $0<k\alpha<\nu$ in~(8.1), and the field $Z^N_n$
is defined by the field $\xi_n=H(X_n)$, $n\in\text{\BBB Z}_\nu$, and
formula~(1.1), then the multi-dimensional distributions of the fields
$Z_n^N$ with $A_N=N^{\nu-k\alpha/2}L(N)^{k/2}$ tend to those of the
fields $c_kZ^*_n$, $n\in\text{\BBB Z}_\nu$, where the field $Z^*_n$ is
the same as in Theorem~8.2.}

\medskip\noindent
{\it Proof of Theorem~$8.2'$.}\/ Define
$H'(x)=\summ_{j=k+1}^\infty c_jH_j(x)$ and
$Y_n^N=\frac1{A_N}\summ_{l\in B_n^N}H'(X_l)$.
Because of Theorem~8.2 in order to prove Theorem~$8.2'$ it is enough
to show that
$$
E(Y_n^N)^2\to0 \quad\text{as } N\to\infty.
$$
It follows from Corollary~5.5 that
$$
E(Y_n^N)^2=\frac1{A_N^2}\sum_{j=k+1}^\infty j!\,
\sum_{s,t\in B_n^N}[r(s-t)]^j.
$$
Hence a simple calculation with the help of formula~(8.1) yields
$$
E(Y_n^N)^2=\frac1{A_N^2}\[O(N^{2\nu-(k+1)\alpha}L(N)^{k+1})+O(N^\nu)\]\to0.
$$
Theorem~$8.2'$ is proved.

\medskip
Let us consider a slightly more general version of the problem 
investigated in Theorem~$8.2'$. Take a stationary Gaussian 
random field  $X_n$, $EX_n=0$, $EX_n^2=1$, $n\in\text{\BBB Z}_\nu$ 
with a correlation  function satisfying relation~(8.1), and the 
field $\xi_n=H(X_n)$, $n\in\text{\BBB Z}_\nu$, subordinated to it 
with a general function $H(x)$ such that $EH(X_n)=0$ and 
$EH(X_n)^2<\infty$. We are interested in the large-scale limit 
of such random fields. Take the Hermite expansion~(8.20) of the 
function~$H(x)$, and let $k$ be the smallest such index for which 
$c_k\neq0$ in the expansion~(8.20). In Theorem~$8.2'$ we solved
this problem if $0<k\alpha<\nu$. We are interested in the 
question what happens in the case when $k\alpha>\nu$. Let 
me remark that in the case $k\alpha\ge\nu$ the field $Z^*_n$, 
$n\in\text{\BBB Z}_\nu$, which appeared in the limit in 
Theorem~$8.2'$ does not exist. The Wiener-It\^o integral
defining $Z^*_n$ is meaningless, because the integral which 
should be finite to guarantee the existence of the Wiener--It\^o
integral is divergent in this case. Next I formulate a general 
result which contains the answer to the above question as a 
special case.

\medskip\noindent
{\bf Theorem~8.5.} {\it Let us consider a stationary Gaussian random
field $X_n$, $EX_n=0$, $EX_n^2=1$, $n\in\text{\BBB Z}_n$, with 
correlation function $r(n)=EX_mX_{m+n}$, $m,n\in\text{\BBB Z}_\nu$.
Take a function $H(x)$ on the real line such that $EH(X_n)=0$ and
$EH(X_n)^2<\infty$. Take the Hermite expansion~(8.20) of the
function~$H(x)$, and let $k$ be smallest index in this expansion
such that $c_k\neq 0$. If 
$$
\sum_{n\in\text{\BBB Z}_\nu}|r(n)|^k<\infty, \tag8.21
$$  
then the limit
$$
\lim_{N\to\infty} EZ^N_n(H_l)^2=\lim_{N\to\infty} 
N^{-\nu}\sum_{i\in B^N_n}\sum_{j\in B^N_n}r^l(i-j)=\sigma_l^2l!
$$
exists for all indices $l\ge k$, where $Z^N_n(H_l)$ is defined 
in~(1.1) with $A_N=N^{\nu/2}$, and $\xi_n=H_l(X_n)$ with 
the $l$-th Hermite polynomial $H_l(x)$ with leading coefficient~1.
Moreover, also the inequality  
$$
\sigma^2=\sum_{l=k}^\infty c_l^2l!\sigma_l^2<\infty
$$
holds. 

The finite dimensional distributions of the random field $Z^N_n(H)$ 
defined  in~(1.1) with $A_N=N^{\nu/2}$ and $\xi_n=H(X_n)$ tend to 
the finite  dimensional distributions of a random field 
$\sigma Z^*_n$ with the  number~$\sigma$ defined in the previous 
relation, where $Z^*_n$, $n\in\text{\BBB Z}_\nu$, are independent, 
standard normal random variables.}

\medskip
Theorem 8.5 can be applied if the conditions of Theorem~$8.2'$ 
hold with the only modification that the condition $k\alpha<\nu$ 
is replaced by the relation $k\alpha>\nu$. In this case the 
relation~(8.21) holds, and the large-scale limit of the random 
field $Z^N_n$, $n\in\text{\BBB Z}_\nu$ with normalization 
$A_N=N^{\nu/2}$ is a random field consisting of independent 
standard normal random variables multiplied with the 
number~$\sigma$. There is a slight generalization of 
Theorem~8.5 which also covers the case $k\alpha=\nu$. In this 
result we assume instead of the condition~(8.21) that 
$\summ_{n\in \bar B_N} r(n)^k=L(N)$ with a slowly varying 
function $L(\cdot)$, where 
$\bar B_N=\{(n_1,\dots,n_\nu)\in\text{\BBB Z}_\nu\colon\;
-N\le n_j\le N,\; 1\le j\le\nu\}$, and some additional 
condition is imposed which states that an appropriately defined 
finite number  $\sigma^2=\limm_{N\to\infty}\sigma_N^2$, which 
plays the role of the variance of the random variables in the 
limiting field, exists. There is a similar large scale limit in 
this case as in Theorem~8.5, the only difference is that the 
norming constant in this case is $A_N=N^{\nu/2}L(N)^{1/2}$. This 
result has the consequence that if the conditions of 
Theorem~$8.2'$ hold with the only differnce that $k\alpha=\nu$ 
instead of $k\alpha<\nu$,  then the large scale limit exists 
with norming constants $A_N=N^{\nu/2}L(N)$ with an  appropriate 
slowly varying function~$L(\cdot)$, and it consists of 
independent Gaussian random variables with expectation zero.

The proof of Theorem~8.5 and its generalization that we did not
formulate here explicitly appeared in paper~[3]. I omit its proof,
I only make some short explanation about it.

In the proof we show that all moments of the random variables 
$Z^N_n$ converge to the corresponding moments of the random 
variables $Z^*_n$  as $N\to\infty$. The moments of the random 
variables $Z_n^N$ can be calculated by means of the diagram 
formula if we either rewrite them in the form of a Wiener--It\^o 
integral or apply a version of the diagram formula which gives 
the moments of Wick polynomials instead of Wiener--It\^o 
integrals. In both cases the moments can be expressed explicitly 
by means of the correlation function of the underlying Gaussian 
random field. The most important step of the proof is to show 
that we can select a special subclass of (closed) diagrams, 
called regular diagrams in~[3] which yield the main contribution 
to the moment $E(Z^N_n)^M$, and their contribution can be 
simply calculated. The contribution of all remaining diagrams 
is~$o(1)$, hence it is negligible. For the sake of simplicity
let us restrict our attention to the case $H(x)=H_k(x)$ when 
defining the regular diagrams. If $M$ is an even number, then 
take a partion $\{k_1,k_2\}$, $\{k_3,k_4\}$,\dots, 
$\{k_{M-1},k_M\}$ of the set $\{1,\dots,M\}$ to subsets 
consisting of exactly two elements, to define the regular
diagrams. They are those (closed) diagrams which contain only 
edges connecting vertices from the $k_{2j-1}$-th and $k_{2j}$-th 
row of the diagram with some $1\le j\le \frac M2$, where 
$\{k_{2j-1},k_{2j}\}$ is an element of the above partition. 
If $M$ is an odd number, then there is no regular diagram. 

\medskip
In Theorems~8.2 and~$8.2'$ we investigated some very special
subordinated fields. The next result shows that  the same 
limiting field as the one in Theorem~8.2 appears in a much 
more general situation.

Let us define the field
$$
\xi_n=\sum_{j=k}^\infty\frac1{j!}\int e^{i(n,x_1+\cdots+x_j)}
\alpha_j(x_1,\dots,x_j)\,Z_G(\,dx_1)\dots Z_G(\,dx_j),
\quad n\in\text{\BBB Z}_\nu, \tag8.22
$$
where $Z_G$ is the random spectral measure adapted to a Gaussian
field~$X_n$, $n\in\text{\BBB Z}_\nu$, with correlation function
satisfying~(8.1) with $0<\alpha<\frac \nu k$.

\medskip\noindent
{\bf Theorem 8.6.} {\it Let the fields $Z^N_n$ be defined by
formulae~(8.22) and~(8.1) with $A_N=N^{\nu-k\alpha/2}$. The
multi-dimensional distributions of the fields $Z_n^N$ tend to
those of the field $\alpha_k(0,\dots,0)Z^*_n$ where the field~$Z^*_n$
is the same as in Theorem~8.2 if the following conditions are
fulfilled:

\medskip
\item{(i)} $\alpha_k(x_1,\dots,x_k)$ is a bounded function, continuous
at the origin, and such that \newline
$\alpha_k(0,\dots,0)\neq0$.
\item{(ii)}
$$
\align
\sum_{j=k=1}^\infty\frac1{j!}\frac{N^{-(j-k)\alpha}}{L(N)^{j-k}}
\int_{ R^{j\nu}}&
\left|\alpha_j\(\frac{x_1}N,\dots,\frac{x_j}N\)\right|^2 \frac1{N^{2\nu}}
\left|\sum_{j\in B_0^N}e^{i(l/N,x_1+\cdots+x_j)}\right|^2 \\
&\qquad G_N(\,dx_1)\dots G_N(\,dx_j)\to0,
\endalign
$$
where $G_N$ is defined in~(8.2).

}

\medskip\noindent
{\it Proof of Theorem 8.6.}\/ The proof is very similar to those of
Theorem~8.2 and~$8.2'$. The same argument as in the proof of
Theorem~$8.2'$ shows that because of condition~(ii) $\xi_n$ can be
substituted in the present proof by the following expression:
$$
\xi_n'=\frac1{k!}\int e^{i(n,x_1+\cdots+x_k)}\alpha_k(x_1,\dots,x_k)
Z_G(\,dx_1)\dots Z_G(\,dx_k), \quad n\in\text{\BBB Z}_\nu.
$$
Then a natural modification in the proof of Theorem~8.2 implies
Theorem~8.6. The main point in this modification is that we have to
substitute the measures~$\mu_N$ defined in formula~(8.11) by the
following measure $\bar\mu_N$:
$$
\bar\mu_N(A)=\int_A|K_N(x_1,\dots,x_k)|^2
\left|\alpha_k\left(\frac {x_1}N,\dots,\frac{x_k}N\right)\right|^2
G_N(\,dx_1)\dots G_N(\,dx_k), \quad A\in\Cal B^{k\nu},
$$
and to observe that because of condition~(i) the limit relation
$\mu_N\overset w\to\rightarrow\mu_0$ implies that
$\bar\mu_N\overset w\to\rightarrow|\alpha_k(0,\dots,0)|^2\mu_0$.

The main problem in applying Theorem~8.6 is to check conditions~(i)
and~(ii). We remark without proof that any field
$\xi_n=H(X_{s_1+n},\dots,X_{s_p+n})$,
$s_1,\dots,s_p\in\text{\BBB Z}_\nu$ and $n\in\text{\BBB Z}_\nu$,
for which $E\xi_n^2<\infty$ satisfies condition~(ii). This is proved
in Remark~6.2 of~[9]. If the conditions~(i) or~(ii) are violated,
then a limit of different type may appear. Finally we quote such a
result without proof. (See~[23] for a proof.) Here we restrict
ourselves to the case $\nu=1$. The limiting field appearing in this
result belongs to the class of self-similar fields constructed in
Remark~6.5.

Let $a_n$, $n=\dots,-1,0,1,\dots$, be a sequence of real numbers 
such that
$$
\aligned
a_n&=C(1)n^{-\beta-1}+o(n^{-\beta-1})\quad \text{if } n\ge 0 \\
a_n&=C(2)|n|^{-\beta-1}+o(|n|^{-\beta-1})\quad \text{if } n< 0
\endaligned \qquad -1<\beta<1. \tag8.22
$$
Let $X_n$, $n=\dots,-1,0,1,\dots$, be a stationary Gaussian 
sequence with correlation function 
$r(n)=EX_0X_n=|n|^{-\alpha}L(|n|)$, \ $0<\alpha<1$, where 
$L(\cdot)$ is a slowly varying function. Define the field 
$\xi_n$, $n=\dots,-1,0,1,\dots$, as
$$
\xi_n=\sum_{m=-\infty}^\infty a_mH_k(X_{m+n}). \tag8.22
$$

\medskip\noindent
{\bf Theorem 8.7.} {\it Let a sequence $\xi_n$, 
$n=\dots,-1,0,1,\dots$, be defined by~(8.23) and~(8.24). Let 
$0<k\alpha<1$, $0<1-\beta-\frac k2\alpha<1$, and let one of 
the following conditions be satisfied.

\medskip
\item{(a)} $0<\beta<1$, and $\summ_{n=-\infty}^\infty a_n=0$.
\item{(b)} $0>\beta>-1$.
\item{(c)} $\beta=0$, $C(1)=-C(2)$, and
$\summ_{n=0}^\infty|a_n+a_{-n}|<\infty$.

\medskip\noindent
Let us define the sequences $Z^N_n$ by formula~(1.1) with
$A_N=N^{1-\beta-k\alpha/2}L(N)^{k/2}$ and the above defined 
field~$\xi_n$. The multi-dimensional distributions of the 
sequences~$Z^N_n$ tend to those of the sequences 
$D^{-k}Z^*_n(\alpha,\beta,a,b,c)$, where
$$
\align
Z^*_n(\alpha,\beta,k,b,c)=\int&\tilde\chi_n(x_1+\cdots+x_k)\\
&\qquad \left[b|x_1+\cdots+x_k|^\beta+ic|x_1+\cdots+x_k|^\beta
\text{\rm sign}\,(x_1+\cdots+x_k)\right] \\
&\qquad\qquad |x_1|^{(\alpha-1)/2}\cdots |x_k|^{(\alpha-1)/2}\,
W(\,dx_1)\dots W(\,dx_k),
\endalign
$$
$W(\cdot)$ denotes the white noise field, i.e. a random spectral
measure corresponding to the Lebesgue measure, and the constants~$D$,
$b$ and~$c$ are defined as $D=2\Gamma(\alpha)\cos(\frac\alpha2\pi)$, and

\medskip
\item{}
$b=2[C(1)+C(2)]\Gamma(-\beta)\sin(\frac{\beta+1}2\pi)$,
$c=2[C(1)-C(2)]\Gamma(-\beta)\cos(\frac{\beta+1}2\pi)$
in case~(a) and~(b), and
\item{} $b=\summ_{n=-\infty}^\infty a_n$, $c=C(1)$
in case~(c).

}

\beginsection 9. History of the problems. Comments.

{\it Section 1.}

\medskip\noindent
In statistical physics the problem formulated in this section appeared
at the investigation of some physical models at critical temperature.
A discussion of this problem and further references can be found in the
fourth chapter of the forthcoming book of Ya.~G.~Sinai~[33]. The
first example of a limit theorem for partial sums of random variables
which is considerably different form the independent case was given by
M.~Rosenblatt in~[28]. Further results in this direction were proved
by R.~L.~Dobrushin, H.~Kesten and F.~Spitzer, P.~Major, M.~Rosenblatt
and M.~S.~Taqqu [7], [8], [9], [23], [29], [30], [34], [37]. In most
of these papers only the one-dimensional case is considered, and it is
formulated in a different but equivalent way. The joint distribution
of the random variables $A_N^{-1}\summ_{j=1}^{Nt]}\xi_j$, $0<t<\infty$,
is considered.

Similar problems also appeared in the theory of infinite particle
systems. The large-scale limit of the so-called voter model and of
infinite particle branching Brownian motions were investigated in
papers [2], [6], [17], [24]. It was proved that in these models the
limit is, with a non-typical normalization, a Gaussian self-similar
field. The investigation of the large-scale limit would be very natural
for many other infinite particle systems, but in most cases this
problem is hopelessly difficult.

The notion of subordinated fields in the present context first appeared
at Dobrushin~[7]. It is natural to expect that there exists a large
class of self-similar fields which cannot be obtained as subordinated
fields. Nevertheless the present techniques are not powerful enough
for finding them.

The approach  to the problem is different in statistical physics. In
statistical physics one looks for self-similar fields which satisfy
some conditions formulated in accordance to physical considerations.
One tries to describe these fields with the help of a power series
which is the Radon--Nykodim derivative of the field with respect to
a Gaussian field. The deepest result in this direction is a recent
paper of P.~M.~Bleher and M.~D.~Missarov~[1] who can define
the required formal power series. This result enables one to
calculate several critical indices interesting for physicists, but
the task of proving that this formal expression defines an existing
field seems to be very hard. It is also an open problem whether the
class of self-similar fields constructed via multiple Wiener--It\^o
integrals contains the non-Gaussian self-similar fields interesting
for statistical physics. Some experts are very skeptical in this
respect. The Gaussian self-similar fields are investigated in~[7]
and~[32]. A more thorough investigation is under preparation in~[11].

The notion of generalized fields was introduced by I.~M.~Gelfand.
A detailed discussion can be found in the book~[15], where the
properties of Schwartz spaces we need can also be found.

In the definition of generalized fields the class of test
functions~$\Cal S$ can be substituted by other linear topological
spaces consisting of real valued functions. The most frequently
considered space, beside the space~$\Cal S$, is the space~$\Cal D$
of infinitely many times differentiable functions with compact
support. In paper~[7] Dobrushin also considered the space
$\Cal S^r\subset\Cal S$, which consists of the functions
$\varphi\in\Cal S$ satisfying the additional relation
$\int {x^{(1)}}^{j_1}\cdots {x^{(\nu)}}^{j_\nu}\varphi(x)\,dx=0$
provided that $j_1+\cdots+j_\nu<r$. He considered this class of
test functions because there are much more continuous linear
functionals over~$\Cal S^r$ than over~$\Cal S$, and this property
of~$\Cal S^r$ can be exploited in certain investigations. Generally
no problem arises in the proofs if the space of test
functions~$\Cal S$ is substituted by~$\Cal S^r$ or $\Cal D$ in the
definition of generalized fields.

Two generalized fields $X(\varphi)$ and $\bar X(\varphi)$ can be
identified if $X(\varphi)\overset\Delta\to=\bar X(\varphi)$ for all
$\varphi\in\Cal S$. Let me remark that this relation also implies
that the multi-dimensional distributions of the random vectors
$(X(\varphi_1),\dots,X(\varphi_n))$ and
$(\bar X(\varphi_1),\dots,\bar X(\varphi_n))$ coincide for all
$\varphi_1,\dots,\varphi_n\in\Cal S$. As $\Cal S$ is a
linear space, this relation can be deduced from property~a) of
generalized fields by exploiting that two distribution functions
on~$R^n$ agree if and only if their characteristic functions agree.

Let $\Cal S'$ denote the space of continuous linear functionals
over~$\Cal S$, and let $\Cal A_{\Cal S'}$ be the $\sigma$-algebra
over $\Cal S'$ generated by the sets
$A(\varphi,a)=\{F\colon\;\; F\in\Cal S';,\,F(\varphi)<a\}$, where
$\varphi\in\Cal S$ and $a\in R^1$ are arbitrary. Given a probability
space  $(\Cal S',\Cal A_{\Cal S'},P)$,  a generalized field
$\bar X=\bar X(\varphi)$ can be defined on it by the formula
$\bar X(\varphi)(F)=F(\varphi)$, \ $\varphi\in\Cal S$, and 
$F\in\Cal S'$. The following deep result is due to Minlos 
(see e.g.~[15]).

\medskip\noindent
{\bf Theorem. (Minlos)} {\it Let $(X(\varphi),\,\varphi\in\Cal S)$ 
be a generalized random field. There exists a probability 
measure~$P$ on the measurable space $(\Cal S,\Cal A_{\Cal S'})$ 
such the generalized field 
$\bar X=(\bar X(\varphi),\,\varphi\in\Cal S)$ defined on the
probability space $(\bar S,\Cal A_{\Cal S'},P)$ by the formula
$\bar X(\varphi)(F)=F(\varphi)$, $\varphi\in\Cal S$, $F\in\Cal S'$,
satisfies the relation $X(\varphi)\overset\Delta\to=\bar X(\varphi)$
for all $\varphi\in\Cal S$.}

\medskip
The generalized field $\bar X$ has some nice properties. Namely
property~a) in the definition of generalized fields holds for all
$F\in\Cal S'$. Moreover $\bar X$ satisfies the following strengthened
version of property~b):

\medskip
\item{b$'$)} $\lim\bar X(\varphi_n)=\bar X(\varphi)$ in every point
$F\in\Cal S'$ if $\varphi_n\to\varphi$ in the topology of~$\Cal S$.

\medskip
Because of this nice behaviour of the field $\bar X(\varphi)$ most
authors define generalized fields as the versions $\bar X$ defined
in Minlos' theorem. Since we have never needed the extra properties
of the field~$\bar X$ we have deliberately avoided the application
of Minlos' theorem in the definition of generalized random fields.
Minlos' theorem heavily depends on some topological properties
of~$\Cal S$, namely that $\Cal S$ is a so-called nuclear space.
Minlos' theorem also holds if the space of test functions is
substituted by $\Cal D$ or $\Cal S^r$ in the definition of
generalized fields.

\medskip
Let us finally remark that Lamperti~[21] gave an interesting
characterization of self-similar random fields. Let $X(t)$, $t\in R^1$,
be a continuous time stationary random process, and define the random
process $Y(t)=\frac{X(\log t)}{t^\alpha}$, $t>0$, with some $\alpha>0$.
Then, as it is  not difficult to see, the random processes $Y(t)$,
$t>0$, and $\frac{Y(ut)}{u^\alpha}$, $t>0$, have the same finite
dimensional distributions for all $u>0$. This can be interpreted so
that $Y(t)$ is a self-similar process with parameter~$\alpha>0$ on
the half-line $t>0$. Contrariwise, if the finite dimensional
distributions of the processes $Y(t)$ and $\frac{Y(ut)}{u^\alpha}$,
$t>0$, agree for all $u>0$, then the process
$X(t)=\frac{X(e^t)}{e^{\alpha t}}$, $t\in R^1$, is stationary.
These relations show some connection between stationary and
self-similar processes. But they have a rather limited importance in
the investigations of this work, because here we are really interested
in such random fields which are simultaneously stationary and
self-similar.

\medskip\noindent
{\it Section 2.}

\medskip\noindent
Wick polynomials are widely used in the literature of statistical 
physics. A detailed discussion about Wick polynomials can be 
found in~[12]. Theorems~2A and~2B are well-known, and they can be 
found in the standard literature. Theorem~2C can be found e.g. in 
Dynkin's book~[13] (Lemma~1.5). Theorem~2.1 is due to Segal~[31]. 
It is closely related to a result of Cameron and Martin~[4]. The 
remarks at the end of the section about the content of formula~2.1 
are related to~[25].

\medskip\noindent
{\it Section 3.}

\medskip\noindent
Random spectral measures were independently introduced by Cramer 
and Kolmogorov [5], [20]. They could have been introduced by 
means of Stone's theorem about the spectral representation of 
one-parameter groups of unitary operators. Bochner's theorem can 
be found in any standard book on functional analysis, the proof 
of the Bochner--Schwartz theorem can be found in~[15]. Let me 
remark that the same result holds true if the space of test 
functions~$\Cal S$ is substituted by~$\Cal D$.

\medskip\noindent
{\it Section 4.}

\medskip\noindent
The stochastic integral defined in this section is a version of that
introduced by~It\^o in~[18]. This modified integral first appeared
in Totoki's lecture note~[38] in a special form. Its definition is a
little bit more difficult than the definition of the original
stochastic integral introduced by It\^o, but it has the advantage
that the effect of the shift transformation can be better studied
with its help. Most results of this section can be found in
Dobrushin's paper~[7]. The definition of Wiener--It\^o integrals
in the case when the spectral measure may have atoms is new. In the
new version of this lecture note I worked out many arguments in a
more detailed form than in the old text. In particular, I have
given a much more detailed explanation of the statement that all
kernel functions of Wiener--It\^o integrals can be well
approximated by elementary functions.

\medskip\noindent
{\it Section 5.}

\medskip\noindent
Proposition~5.1 is proved for the original Wiener--It\^o integrals
by It\^o in~[18]. Lemma~5.2 contains a well-known formula about
Hermite  polynomials. The main result of this section, Theorem~5.3,
appeared in Dobrushin's work~[7]. The proof given there is not
complete. Several non-trivial details are omitted. I felt even
necessary to present a more detailed proof in this note when I
wrote down its new version. Theorem~5.3 is closely related to
Feynman's diagram formula. The result of Corollary~5.5 was already
known at the beginning of this century. It was proved with the help
of some formal manipulations. This formal calculation was justified
by Taqqu in~[35] with the help of some deep inequalities. In the
new version of this note I formulated a more general result than in
the older one. Here I gave a formula about the moment of products
of Wick polynomials and not only of Hermite polynomials.

I could not find results similar to Propositions~5.6 and~5.7 in the
literature of probability theory. On the other hand, such results are
well-known in statistical physics, and they play an important role
in constructive field theory. A sharpened form of these results is
Nelson's deep hypercontractive inequality~[27], which I formulate
below.

Let $X_t$, $t\in T$, and $Y_{t'}$, $t'\in T'$ be two sets of jointly
Gaussian random variables on some probability spaces $(\Omega,\Cal A,P)$
and $(\Omega,\Cal A',P')$. Let $\Cal H_1$ and $\Cal H_1'$ be the
Hilbert spaces generated by the finite linear combinations
$\sum c_jX_{t_j}$ and $\sum c_jY_{t'_j}$. Let us define the
$\sigma$-algebras $\Cal B=\sigma(X_t,\,t\in T)$ and
$\Cal B'=\sigma(Y_{t'},\,t'\in T')$ and the Banach spaces
$L_p(X)=L_p(\Omega,\Cal B,P)$,
$L_p(Y)=L_p(\Omega',\Cal B',P')$, $1\le p\le\infty$. Let $A$ be
linear transformation from $\Cal H_1$ to $\Cal H_1'$ with norm not
exceeding~1. We define an operator $\Gamma(A)\colon L_p(X)\to L_{p'}(Y)$
for all $1\le p,p'\le\infty$ in the following way. If $\eta$ is a
homogeneous polynomial of the variables~$X_t$,
$$
\eta=\sum C_{j_1,\dots,j_s}^{t_1,\dots,t_s} X^{j_1}_{t_1}\cdots X^{j_s}_{t_s},
\quad t_1,\dots,t_s\in T,
$$
then
$$
\Gamma(A)\:\!\eta\!\:=\sum C_{j_1,\dots,j_s}^{t_1,\dots,t_s}
\:\!(AX_{t_1})^{j_1}\cdots (AX_{t_s})^{j_s}\!\:.
$$
It can be proved that this definition is meaningful, i.e.
$\Gamma(A)\:\!\eta\!\:$ does not depend on the representation of $\eta$,
and $\Gamma(A)$ can be extended to a bounded operator from $L_1(X)$
to $L_1(Y)$ in a unique way. This means in particular that $\Gamma(A)\xi$
is defined for all $\xi\in L_p(X)$, $p\ge1$. Nelson's hypercontractive
inequality says the following. Let $A$ be a contraction from $\Cal H_1$
to $\Cal H_1'$. Then $\Gamma(A)$ is a contraction from $L_q(X)$ to
$L_p(Y)$ for $1\le q\le p$ provided that
$$
\|A\|\le\( \frac{q-1}{p-1}\)^{1/2}. \tag$+$
$$
If $(+)$ does not hold, then $\Gamma(A)$ is not a bounded operator from
$L_q(X)$ to $L_p(Y)$.

A further generalization of this result can be found in~[16].

The following discussion may help to understand the relation between
Nelson's hypercontractive inequality and Corollary~5.6. Let us apply
Nelson's inequality in the special case when
$(X_t,\,t\in T)=(Y_{t'},\,t'\in T')$ is a stationary Gaussian field with
spectral measure~$G$, $q=2$, $p=2m$ with some positive integer~$m$,
$A=c\cdot\text{Id}$, where $\text{Id}$ denotes the identity operator,
and $c=(2m-1)^{-1/2}$. Let $\Cal H^c$ and $\Cal H^c_n$ be the
complexification of the real Hilbert spaces $\Cal H$ and $\Cal H_n$
defined in Section~2.
Then $L_2(X)=\Cal H^c=\Cal H^c_0+\Cal H_1^c+\cdots$ by Theorem~2.1 and
formula~2.1. The operator $\Gamma(c\cdot\text{Id}\,)$ equals
$c^n\cdot\text{Id}$ on the subspace $\Cal H_n^2$. If $h_n\in\Cal H^n_G$,
then $I_G(h_n)\in \Cal H_n$, hence the application of Nelson's
inequality for the operator $A=c\cdot\text{Id}$ shows that
$$
\(EI_G(h_n)^{2m}\)^{1/2m}
=c^{-n}\(E(\Gamma(c\cdot\text{Id})I_G(h_n))^{2m}\)^{1/2m}
\le c^{-n}\(EI_G(h_n)^2\)^{1/2}
$$
i.e.
$$
EI_G(h_n)^{2m}\le c^{-2nm}\(EI_G(h_n)^2\)^m=(2m-1)^{mn}\(EI_G(h_n)^2\)^m.
$$
This inequality is very similar to the second inequality in
Corollary~5.6, only the multiplying constants are different. Moreover,
for large~$m$ these multiplying constants are near to each other. I
remark that the following weakened form of Nelson's inequality could
be deduced relatively easily from Corollary~5.6. Let
$A\colon\;\Cal H_1\to\Cal H'_1$ be a contraction $\|A\|=c<1$. Then
there exists a $\bar p=\bar p(c)>2$ such that $\Gamma(A)$ is a
bounded operator from $L_2(X)$ to $L_p(Y)$ for $p<\bar p$. This
weakened form of Nelson's inequality is sufficient in many
applications.

\medskip\noindent
{\it Section 6.}

\medskip\noindent
Theorems~6.1,~6.2 and Corollary~6.4 were proved by Dobrushin in~[7]. 
Taqqu proved similar results in~[36], but he gave a different 
representation. Theorem~6.6 was proved by H.~P.~Mc.Kean in~[26]. 
The proof of the lower bound uses some ideas from~[14]. Remark~6.5 
is from~[23]. As Proposition~6.3 also indicates, some non-trivial 
problems about the convergence of certain integrals must be solved 
when constructing self-similar fields. Such convergence problems are 
common in statistical physics. To tackle such problems the 
so-called power counting method (see e.g.~[22]) was worked out. 
This method could also be applied in this section. Part~b) of 
Proposition~6.3 implies that the self-similarity parameter~$\alpha$ 
cannot be chosen in a larger domain in Corollary~6.4. One can ask 
about the behaviour of the random variables $\xi_j$ and 
$\xi(\varphi)$ defined in Corollary~6.4 if the self-similarity
parameter~$\alpha$ tends to the critical value~$\frac\nu2$. The 
variance of the random variables~$\xi_j$ and $\xi(\varphi)$ tends 
to infinity in this case, and the fields $\xi_j$, 
$j\in\text{\BBB Z}_\nu$, and $\xi(\varphi)$, $\varphi\in\Cal S$, 
tend, after an appropriate renormalization, to a field of 
independent normal random variables in the discrete, and to a white 
noise in the continuous case. The proof of these results with a 
more detailed discussion will appear in~[10].

In a recent paper~[19] Kesten and Spitzer have proved a limit theorem,
where the limit field is a self-similar field which seems not to belong
to the class of self-similar fields constructed in Section~6. (We
cannot however, exclude the possibility that there exists some
self-similar field in the class defined in Theorem~6.2 with the
same distribution as this field, although it is given by a completely
different form.) This self-similar field constructed by Kesten and
Spitzer is the only rigorously constructed self-similar field known
for us that does not belong to the fields constructed in Theorem~6.2.
I describe this field, and then I make some comments.

Let $B_1(t)$ and $B_2(t)$, $-\infty<t<\infty$, be two independent
Wiener processes. (We say that $B(t)$ is a Wiener process on the
real line if $B(t)$, $t\ge0$, and $B(-t)$, $t\ge0$, are two independent
Wiener processes.) Let $K(x,t_1,t_2)$, $x\in R^1$, $t_1<t_2$, denote
the local time of the process $B_1$ at the point $x$ in the interval
$[t_1,t_2]$. The one-dimensional field
$$
Z_n=\int K(x,n,n+1)\,B_2(\,dx), \quad n=\dots,-1,0,1,\dots,
$$
where the integral in the last formula is an It\^o integral, is a
stationary self-similar field with self-similarity parameter~$\frac34$.

To see the self-similarity property one has to observe that
$$
K(\lambda^{1/2}x,\lambda t_1,\lambda t_2)\overset\Delta\to=\lambda^{1/2}
K(x,t_1,t_2) \quad\text{for all }x\in R^1,\;\; t_1<t_2, \text{ and }
\lambda>0
$$
because of the relation
$B_1(\lambda u)\overset\Delta\to=\lambda^{1/2}B_1(u)$. Hence
$$
\sum_{j=0}^{n-1}Z_j\overset\Delta\to=n^{1/2}\int K(n^{-1/2}x,0,1)B_2(\,dx)
\overset\Delta\to= n^{3/4}\int K(x,0,1)\,B_2(\,dx)=n^{3/4}Z_0.
$$
The invariance of the multi-dimensional distributions of the field~$Z_n$
under the transformation~(1.1) can be seen similarly.

To see the stationarity of the field~$Z_n$ we need the following two
observations.

\medskip
\item{a)}$ K(x,s,t)\overset\Delta\to=K(x+\eta(s),0,t-s)$ with
$\eta(s)=-B_1(-s)$. (The form of $\eta$ is not important for us. What
we need is that the pair $(\eta,K)$ is independent of $B_2$.)
\item{b)} If $\alpha(x)$, $-\infty<x<\infty$, is a process independent
of $B_2$, then
$$
\int \alpha(x+u)B_2(\,dx)\overset\Delta\to=\int\alpha(x) B_2(\,dx)
\quad\text{for all }u\in R^1.
$$

\medskip
It is enough to show, because of Property~a) that
$$
\int K(x+\eta(s),0,t-s)\,B_2(\,dx)\overset\Delta\to=
\int K(x,0,t-s)\,B_2(\,dx).
$$
This relation follows from property~b), because the conditional
distributions of the left and right-hand sides agree under the
condition $\eta(s)=u$, $u\in R^1$.

The generalized field version of the above field~$Z_n$ is the field
$$
Z(\varphi)=-\int \left[K(x,0,t)\frac{d\varphi}{dt}\,dt\right] B_2(\,dx),
\quad \varphi\in\Cal S.
$$
To explain the analogy between the field $Z_n$ and $Z(\varphi)$ we remark
that the kernel of the integral defining~$Z_n$ can be written, at
least formally, as
$$
K(x,n,n+1)=\int\chi_{[n,n+1)}(u)\frac d{du}K(x,n,u)\,du,
$$
although $K$ is a non-differentiable function. Substituting the
function $\chi_{[n,n+1)}$ by $\varphi\in\Cal S$, and integrating
by parts (or precisely, considering $\frac d{du}K$ as the derivative
of a distribution) we get the above definition of $Z(\varphi)$.

Using the same idea as before, a more general class of self-similar
fields can be constructed. The integrand $K(x,n,n+1)$ can be substituted
by the local time of any self-similar field with stationary increments
which is independent of $B_2$. Naturally, it must be clarified first
that this local time really exists. One could enlarge this class
also by integrating with respect to a self-similar field with
stationary increments, independent of~$B_1$. The integral with
respect to a field independent of the field $K(x,s,t)$ can be defined
without any difficulty.

\medskip
There seems to be no natural way to represent the above random fields
as random fields subordinated to a Gaussian random field. On the other
hand, the local times $K(x,s,t)$ are measurable with respect to~$B_1$,
they have finite second moments, therefore they can be expressed by means
of multiple Wiener--It\^o integrals with respect to a white noise field.
Then the process $Z_n$ itself can also be represented via multiple
Wiener--It\^o integrals. It would be interesting to know whether the
above defined self-similar fields, and probably a larger class of
self-similar fields, can be constructed in a simple natural way via
multiple Wiener--It\^o integrals with the help of a randomization.

\medskip\noindent
{\it Section 7.}

\medskip\noindent
The definition of Wiener--It\^o integrals together with the proof
of Theorem~7.1 and Proposition~7.3 are given by It\^o in~[18].
Theorem~7.2 is proved in Taqqu's paper~[37]. He needed this result
to show that the self-similar fields defined in~[9] by means of
Wiener--It\^o integrals coincide with the self-similar fields
defined in~[37] by means of modified Wiener--It\^o integrals.

\medskip\noindent
{\it Section 8.}

\medskip\noindent
The results of this section, with the exception of Theorem~8.7 are
proved in~[9]. Theorem~8.7 is proved in~[23]. This paper was strongly
motivated by ~[29]. Lemma~8.3 is formulated in a slightly more
general form than Lemma~3 in~[9]. The present formulation is more
complicated, but it is more useful in some applications. Let me
explain this in more detail. The difference between the original and
the present formulation of this lemma is that here we allow that the
integrand~$K_0$ in the limiting stochastic integral is discontinuous
on a small subset of~$R^{k\nu}$, and the functions~$K_N$ may not
converge on this set. This freedom can be exploited in some
applications. Indeed, let us consider e.g.\ the self-similar fields
constructed in Remark~6.5. In case $p<0$ the integrand in the formula
expressing these fields is not continuous on the hyperplane
$x_1+\cdots+x_n=0$. Hence, if we want to prove limit theorems where
these fields appear as the limit, and this happens e.g. in
Theorem~8.7 then we can apply Lemma~8.3, but not its original version
Lemma~3 in~[9].

The example for non-central limit theorems given by Rosenblatt 
in~[28] and its generalization by Taqqu in~[34] are special cases of
Theorem~8.2. In these papers only the special case $H_k(x)=x^2-1$
is considered. Later Taqqu~[37] proved a result similar to
Theorem~$8.2'$, but he needed more restrictive conditions. The
observation that Theorem~$8.2'$ can be deduced from Theorem~8.2 is
from Taqqu~[34].

The method of~[28] and~[34] does not apply for the proof of Theorem~8.2
in the case of $H_k(x)$, $k\ge3$.  In these papers it is proved that
the moments of the random variables $Z_n^N$ converge to the
corresponding moments of $Z^*_n$. (Actually a different but equivalent
statement is established in these papers.) This convergence of the
moments implies the convergence
$Z_n^N\overset{\Cal D}\to\rightarrow Z^*_n$ if and only if the
distribution of $Z^*_n$ is uniquely determined by its moments.

Theorem~6.6 implies that the $n$-th moment of a $k$-fold Wiener--It\^o
integral equals to $e^{(kn\log n)/2+O(n)}$. Hence some results about
the so-called moment problem show that the distribution of a $k$-fold
Wiener--It\^o integral is determined by its moments only for $k=1$
and $k=2$. Therefore the method of moments does not work in the
proof of Theorem~8.2 for $H_k(x)$, $k\ge3$.

Throughout Section~8 I have assumed that the correlation function of
the underlying Gaussian field to which our fields are subordinated
satisfies formula~(8.1). This assumption seems natural, since it
implies that the spectral measure of the Gaussian field satisfies
Lemma~8.1, and such a condition is needed when $Z_{G_N}$ is substituted
by~$Z_{G_0}$ in the limit. It can be asked whether in Theorem~8.2
formula~8.1 can be substituted by the weaker assumption that the
spectral measure of the Gaussian field satisfies Lemma~8.1. This
question was  investigated in Section~4 of~[9]. The investigation of
the moments shows that the answer is negative. The reason for it is
that the validity of Lemma~8.1, unlike that of Theorem~8.2, does not
depend on whether the spectral measure~$G$ has large singularities
outside the origin or not. The discussion in~[9] also shows that
the Gaussian case, that is the case when $H_k(x)=H_1(x)=x$ in
Theorem~8.2, is considerably different from the non-Gaussian case. A
forthcoming paper of M.~Rosenblatt~[30] gives a better insight into
the above question.

The limiting fields appearing in Theorem~8.2 and~8.6 belong to a 
special subclass of the self-similar fields defined in Theorem~6.2. 
These results  indicate that the self-similar fields defined in 
formula~(6.5) have a much greater range of attraction if the 
homogeneous function~$f_n$ in~(6.5) is the constant function. The 
reason for the particular behaviour of these fields is that the 
constant function is analytic, while a general homogeneous 
function typically has a singularity at the origin. A more 
detailed discussion about this problem can be found in~[23].

\beginsection References.

\medskip

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\vfill\eject

\noindent
{\it Subject index.}

\medskip

Bochner's theorem p.13, 19--20

Bochner Schwartz theorem p.13, 21

canonical representation of subordinated fields p.56

complete diagram p.48

convergence of generalized fields in distribution p.3

diagram p.38

diagram formula p.38

discrete random fields p.1

Fock space p.23, 65

formula for change of variables for Wiener--It\^o integrals p.32

generalized random field (stationary, Gaussian) p.3

Hermite polynomial p.8

It\^o's formula p.31, 66-67

Karamata's theorem p.69

Minlos' theorem p.90

modified Fourier transform p.78

Nelson's hypercontractive inequality p.92

positive definite function p.18, 19

random orthogonal measure p.64

random spectral measure

corresponding to a spectral measure p.14
\hskip1truecm adapted to a random field p.15--16

regular system (and function adapted to it) p.23

Schwartz spaces p.3, 5

self-similar field, self-similarity parameter

for discrete field p.1
\hskip1truecm for generalized field p.4

shift transformation p.2, 4, 9

slowly varying function p.69

spectral measure p.12

stochastic integral (one-fold) p.15

subordinated field

discrete p.2
\hskip1truecm generalized p.4

vague convergence p.69

weak convergence p.77

Wick polynomials p.9

Wiener--It\^o integral p.23--28, 33, 66

white noise p.67

\vfill\eject

\centerline{\it Notations.}

\parindent=0pt

\medskip\medskip\medskip

$\Cal A_{\Cal S'}$ \ p.90

$\Cal D$ \ p.7; \
$\Cal D'$ \ p.7

$\text{Exp}\,\Cal H_G$ \ p.22; \
$\text{Exp}\,\Cal K_\mu$ \ p.65

$f\underset k\to\times g$ \ p.36

$\Cal H$ \ p.8; \
$H_n(x)$ \ p.8; \
$\Cal H_1$ \ p.8; \
$\Cal H_n$ \ p.9; \
$\Cal H_{\le n}$  p.9; \
$\Cal H_{\le n}(\xi_1,\dots,\xi_n)$ \ p.9; \
$\Cal H_1^c$, \ p.13; \
$\bar{\Cal H}_G^n$ \ p.23; \
$\Cal H_G^n$ \ p.23; \
$\hat{\bar{\Cal H}}_G^n$ \ p.23; \
$\hat{\Cal H}_G^n$ \ p.24; \
$h_\gamma(x_1,\dots,x_{N-2|\gamma|})$ \ p.38; \
$h_\gamma$ \ p.38

$I(\cdot)$ \ p.13; \
$I_G(f)$ \ p.23; \
$I_G(f_n)$ \ p.23

$\Cal K_\mu$ \ p.65; \
$\Cal K_\mu^n$ \ p.65; \
$\bar{\Cal K}_\mu^n$ \ p.65; \
$\hat{\bar{\Cal K}}_\mu^n$ \ p.65; \
$\Cal K_n$ \ p.66;
$\Cal K_{\le n}$ \ p.66

$L_G^2$ \ p.13

$\:\!P(\xi_1,\dots,\xi_n)\!\:$ \ p.9

$\Cal S$ \ p.3; \
$\Cal S^c$ \ p.5; \
$\Cal S'$ \ p.5; \
$\Cal S_\nu$ \ p.3; \
$\Cal S^r$ \ p.90; \
$S_{\nu-1}$ \ p.57; \
$\text{Sym}\,f$ \ p.22

$T_m$ \ p.2

$X^A_t(\varphi)$ \ p.3; \
$X(\varphi)$ \ p.3

$\text{\BBB Z}_\nu$ \ p.1; \
$Z_G$ \ p.14; \
$Z(\,dx)$ \ p.49

$\Gamma(n_1,\dots,n_m)$ \ p.38; \
$\bar\Gamma(k_1,\dots,k_p)$ \ p.48; \
$\bar\Gamma$ \ p.48; \
$\tilde\chi_n(x)$ \ p.52; \
$\Pi_n$ \ p.22

$\overset {\Cal D}\to\rightarrow$ \ p.3; \
$\overset v\to\rightarrow$ \ p.69; \
$\overset w\to\rightarrow$ \ p.77; \
$\overset\Delta\to=$ \ p.1; \
$\ominus$ \ p.9; \
$\tilde{}$ \ p.12; \
$*$ \ p.13; \
$\int'$ \ p.66; \
$[\cdot]$ \ p.78

\bye
