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\title[Wiener--It\^o integrals in Gaussian stationary random fields  II]
{The theory of Wiener--It\^o integrals in vector-valued
Gaussian stationary random fields. Part II}

\author[P\'eter Major]{P\'eter Major}

\address{Alfr\'ed R\'enyi Institute of Mathematics  
Budapest, P.O.B. 127 H--1364, Hungary} 
\email{\tt major@renyi.hu}

\keywords{Multiple Wiener--It\^o integrals, multivariate version 
of It\^o's formula, Wick polynomials, shift transformation, 
vague convergence of complex measures, non-central limit theorems.}

\subjclass[2010]{60G10, 60G15, 60F99}

\begin{document}

\begin{abstract}
This work is the continuation of my paper in Moscow
Math. Journal Vol. 20, No. 4 in 2020. In that paper first
I proved the existence of the spectral measure of a
vector-valued stationary Gaussian random field. I also
constructed there the vector-valued random spectral measure
corresponding to this spectral measure, presented its most 
important properties, and defined the multiple Wiener--It\^o 
integrals with respect to it. I also proved an important identity
about the products of multiple Wiener--It\^o integrals which is
called the diagram formula. Here I prove with the help of these
results the multivariate version of It\^o's formula which
shows a relation between multiple Wiener--It\^o integrals with
respect to vector-valued random spectral measures and Wick 
polynomials. Wick polynomials are the multivariate versions of 
Hermite polynomials. I also prove a formula that expresses 
the shift transforms of a random variable given in the form of 
a multiple Wiener--It\^o integral. These results enable us to 
rewrite certain non-linear functionals of a vector-valued 
stationary Gaussian random field in a useful form which suggests 
a limiting procedure that leads to interesting limit theorems.
Finally, I show when this limiting procedure may be carried out,
i.e., when the limit theorems suggested by our new representation
of the investigated non-linear functionals are valid. 
\end{abstract}

\maketitle

\section{Introduction. The main results of the paper.}

This work deals with the properties of vector-valued stationary
Gaussian random fields. In particular, one of its subjects is the
problem how to prove non-central limit theorems for certain non-linear
functionals of such random fields. It is the continuation of 
paper~\cite{8}. These two papers deal with a generalized version
of the problems studied in~\cite{6}. In Lecture Note~\cite{6} 
I investigated scalar-valued stationary random fields. I gave
a good representation of non-linear functionals of a stationary
Gaussian random field with the help of  multiple Wiener--It\^o
integrals with respect to the random spectral measure of this
Gaussian random field. The definition of random spectral
measures and multiple Wiener--It\^o integrals with respect to
it is explained in this Lecture Note. They made possible to prove
some interesting non-central limit theorems.

This work together with paper~\cite{8} contain the natural
multivariate version of the results in~\cite{6}. Their results
provide a useful representation of non-linear functionals of
a vector-valued stationary Gaussian random field with the help
of (vector-valued) multiple Wiener--It\^o integrals with respect
to the random spectral measure of the underlying Gaussian field.
The notions needed to understand the meaning of these results
are also introduced in these works. Some useful properties of
the multiple Wiener--It\^o integrals are also proved. In a
subsequent paper~\cite{9} I will prove the natural multivariate
version of the non-central limit theorems formulated
in~\cite{6} with
the help of these results.

In Theorem~6 of his paper~\cite{1} Arcones formulated the
multivariate version of the non-central limit theorems
proved in~\cite{6}. Actually, this is the result we want
to prove in~\cite{9}. But the formulation and proof of 
Arcones' results are unsatisfactory. He assumed that the
natural multivariate versions of some results in~\cite{6}
hold true. But he did not formulate precisely the results
he applied, and even the definitions of the notions needed
in the formulation of his results are missing. In this
work and in~\cite{8} I fill this gap.

\medskip
Next, I briefly describe the content of the present work
and compare it with the results of~\cite{6}. In this
description I also mention some results in~\cite{8}.

In paper~\cite{8} I have worked out the theory of
vector-valued stationary Gaussian random fields. I introduced
the spectral measure and the random spectral measure of a
vector-valued stationary Gaussian random field. I also defined
a class of generalized stationary Gaussian fields together
with their spectral and random spectral measures. The
subsequent results in~\cite{8} are valid both for classical
and for these new generalized stationary Gaussian random
fields. I defined multiple Wiener--It\^o integrals with
respect to random spectral measures and proved some
important results about them. In particular, I proved the
so-called diagram formula which enables us to rewrite the
product of multiple Wiener--It\^o integrals in the form
of a sum of appropriately defined multiple Wiener--it\^o
integrals. The work~\cite{6} contained the one-dimensional
version of these results.

In~\cite{6} I considered a scalar-valued stationary
Gaussian random field $X(p)$, $p\in\mathbb Z^\nu$, on the 
$\nu$-dimensional integer lattice of the Euclidean 
space $\mathbb R^\nu$, with expectation $EX(p)=0$. I have 
introduced the Hilbert space ${\mathcal H}$ consisting of those 
random variables with finite second moment which are measurable 
with respect to the $\sigma$-algebra generated by the random 
variables $X(p)$, $p\in\mathbb Z^\nu$, of our random field. Here
the usual scalar product $\langle\xi,\eta\rangle=E\xi\eta$ is
applied, and our goal is to understand the behavior of this
Hilbert space.

It is proved in \cite{6} that this Hilbert space ${\mathcal H}$
has a natural representation as the direct sum 
${\mathcal H}={\mathcal H}_0+{\mathcal H}_1+{\mathcal H}_2+\cdots$ 
of orthogonal subspaces ${\mathcal H}_n$, $0\le n<\infty$, which 
are invariant subspaces of the shift transforms in the underlying 
stationary Gaussian random field, and the subspace ${\mathcal H}_n$
consists of those random variables which can be written in the
form of an $n$-fold Wiener--It\^o integral with respect to the
random spectral measure of the underlying Gaussian random field.
(Actually,~\cite{6} gives a more detailed description of the
structure of the space ${\mathcal H}$ with the help of the
so-called Fock space representation. The definition of  the
Fock space is given on page 28 of~\cite{6}, and it is denoted
by $\textrm{Exp}\,{\mathcal H}_G$ there.) The relation between
the Fock space and the Hilbert space ${\mathcal H}$ is
formulated in Theorem~4.2 of~\cite{6} with the help of this
notation. Here I do not discuss this result.) The proof of the
above representation of the Hilbert space ${\mathcal H}$
is based on the so-called It\^o formula which shows a relation
between Hermite polynomials and multiple Wiener--It\^o integrals.
Another important result in~\cite{6}, whose proof is also based
on It\^o's formula is a useful formula that enables us to 
calculate the shift transforms of a random variable presented 
in the form of a multiple Wiener--It\^o integral.

Section~2 of this paper deals with the generalization of the above 
results if we are working with vector-valued stationary Gaussian
random fields. First, the formulation and proof of the multivariate 
version of It\^o's formula are given. In this result, the Wick
polynomials, the multivariate generalizations of Hermite polynomials,
take the role of Hermite polynomials. At the start of Section~2, I
recall their definition and most important properties from~\cite{6}.
Then I prove the multivariate version of It\^o's formula by
adapting the method of proof of its one-dimensional version.
I discuss the proof of this result in Appendix~A of this paper.
I explain there the picture behind the definition of Wick
polynomials, about the idea of the proof of the It\^o's formula, and
why Wick polynomials appear in It\^o's formula when vector-valued
Gaussian random fields are considered. I also discuss the question
how the remaining results of Section~2 are proved with the help of
It\^o's formula.

In this paper I consider vector-valued stationary Gaussian 
random fields  $X(p)=(X_1(p),\dots,X_d(p))$, 
$p\in{\mathbb Z}^\nu$, with expectation $EX(p)=0$,
$p\in{\mathbb Z}^\nu$, and a Hilbert space ${\mathcal H}$ 
defined similarly in the case of scalar-valued random
fields. It is the Hilbert space consisting of those random 
variables with finite second moment which are measurable 
with respect to the $\sigma$-algebra generated by the 
random vectors $X(p)$, $p\in\mathbb Z^\nu$, of our random 
field. Similarly to the one-dimensional case, there is a
decomposition of the Hilbert space ${\mathcal H}$ to the
direct product    
${\mathcal H}={\mathcal H}_0+{\mathcal H}_1+{\mathcal H}_2+\cdots$ 
of orthogonal subspaces, ${\mathcal H}_n$, $0\le n<\infty$, which 
are invariant subspaces of the shift transformations in the 
underlying stationary random field. But in the case of
vector-valued stationary Gaussian random fields we can prove
only a weaker result about the behavior of the subspaces
${\mathcal H}_n$ than in the scalar-valued case. It is
proved in Proposition~2.3 that the elements of an everywhere
dense linear subspace of ${\mathcal H}_n$ can be written in
the form of a finite sum of multiple Wiener--It\^o integrals of
order~$n$. More explicitly, the Wick polynomials of order~$n$
can be written as a finite sum of multiple Wiener--It\^o integrals
of order~$n$, and they constitute an everywhere dense linear
subspace of ${\mathcal H}_n$. On the other hand, we cannot 
represent all elements of ${\mathcal H}_n$ in such a way.
But even this weaker result is sufficient for our purposes.

The last result of Section~2, Proposition~2.4, contains a 
useful formula about the calculation of the shift transforms 
of a random variable given in the form of a multiple 
Wiener--It\^o integral. This formula is similar to the 
analogous result in the case of scalar-valued 
stationary random fields.

In Section~3 we deal with the question how the
previously proved results can be applied in the
investigation of limit theorems for non-linear
functionals of vector-valued stationary Gaussian
random fields.

In several interesting cases (and the problem investigated 
in paper~\cite{9} belongs to them) the limit problem
we are interested in can be reformulated with the help of
It\^o's formula and an appropriate rescaling of certain
multiple Wiener--It\^o integrals we are working with to the
study of a sequence of random variables presented in a very 
special form, and in Section~3 we are investigating
limit theorems for such sequences of random variables.

We consider a sequence of random variables $Z_N$, 
$N=1,2,\dots$, presented as a finite sum of $k$-fold 
Wiener--It\^o integrals with respect to a $d$-dimensional 
random spectral measure. We are interested in the behavior
of such a sequeqnce of random variables $Z_N$, $N=1,2,\dots$,
whose elements are defined by formula~(\ref{3.7}) with
the help of random spectral measures
$Z_{G^{(N)}}=(Z_{G^{(N)},1},\dots,Z_{G^{(N)},d})$  
which correspond to some spectral measures                 
$G^{(N)}=(G^{(N)}_{j,j'})$, $1\le j,j'\le d$, defined on some 
torus $[-A_N,A_N)^\nu$, where $A_N\to\infty$ as $N\to\infty$, 
and we integrate some kernel functions 
$h^{(N)}_{j_1,\dots,j_n}(x_1,\dots,x_d)$ with respect to them.
Let us remark that the random variables $Z_N$ introduced
in~(\ref{3.7}) are sums of finitely many multiple Wiener--It\^o
integrals. Each element of this sum is indexed by some
vector $(j_1,\dots,j_k)$, and the set of these indices does
not depend on the parameter~$N$. We expect that if both
the spectral measures $G^{(N)}$ and the kernel functions
$h^N_{(j_1,\dots,j_n)}$ appearing in the definition of the
random variables $Z_N$ behave nicely, then these random
variables have a limit as $N\to\infty$. Somewhat more
explicitly, we expect that if the spectral measures
$G^{(N)}$ converge to a spectral measure $G^{(0)}$ of a
$d$-dimensional (generalized) stationary Gaussian
random field, and the kernel functions
$h^N_{(j_1,\dots,j_k)}(x_1,\dots,x_d)$ converge to some
nice functions $h^0_{(j_1,\dots,j_k)}(x_1,\dots,x_d)$ as
$N\to\infty$, then the random variables $Z_N$ converge in
distribution to the random variable $Z_0$ defined
in~(\ref{3.10}) with the help of multiple Wiener--It\^o
integrals of the kernel functions $h^0_{(j_1,\dots,j_k)}(x_1,\dots,x_d)$ 
with respect to the random spectral measure $Z_{G^{(0)}}$
corresponding to the spectral measure $G^{(0)}$. Naturally,
in the formulation of such a result we have to clarify
what kind of limit should hold for the spectral
measures $G^{(N)}$ and for the kernel functions
$h^N_{(j_1,\dots,j_k)}(x_1,\dots,x_d)$.

The main result of Section 3 is Proposition 3.1. First the
random variables $Z_N$, $N=0,1,2,\dots$, mentioned in
the above discussion are precisely defined in
formulas~(\ref{3.7}) and ~(\ref{3.10}), and then
Proposition~3.1 yields a sufficient condition for the weak
convergence of the random variables $Z_N$ to $Z_0$ as
$N\to\infty$. This condition consists of two parts. The
first part of them demands that the spectral measures
$G^{(N)}$ converge to the spectral measure $G^{(0)}$ and the
kernel functions $h^N_{(j_1,\dots,j_k)}(x_1,\dots,x_d)$
converge to $h^0_{(j_1,\dots,j_k)}(x_1,\dots,x_d)$ as
$N\to\infty$ in a right way. This is the condition
suggested by the heurisitic argument of the previous
paragraph. But to satisfy the desired weak convergence
we have to impose another condition formulated in condition~(b)
of Proposition~3.1. This is a compactness type condition.
Heuristically, it has the following content. There is such
a compact set in ${\mathbb R}^{k\nu}$ that all multiple
Wiener--It\^o integrals appearing in the definition
of one of the random variables $Z_N$, $N=1,2,\dots$ are
essentially concentrated in this compact set. In the Appendix
I explain the role of this condition with the help of some
examples.

The condition about the limiting behavior of the kernel
functions  \hfill\break
$h^N_{(j_1,\dots,j_k)}(x_1,\dots,x_d)$ is formulated
in condition~(a) of Proposition~3.1. The condition imposed
on the convergence of the spectral measures $G^{(N)}$
deserves special attention.

We are working with spectral measures $(G^{(N)})=(G^{(N)}_{j,j'})$,
$N=0,1,2,\dots$, $1\le j,j'\le d$, and we impose such a condition
that for all indices $1\le j,j'\le d$ the coordinates $G_{j,j'}^{(N)}$
of our spectral measures converge to the coordinate $G_{j,j'}^{(0)}$
as $N\to\infty$ in an appropriate way. We want to consider such
cases when $G^{(0)}$ is the spectral measure of such a generalized
stationary Gaussian random field which belongs to the class of
generalized stationary Gaussian random fields constructed in
Section~4 of~\cite{8}. This must be taken into consideration
when  we want to decide what kind of convergence of the spectal
measures we impose in Proposition~3.1. The coordinates of the
spectral measures of the generalized stationary Gaussian random
fields we are working with are complex measures on ${\mathbb R}^\nu$
with locally finite total variation. (The definition of this
notion is explained in Section~4 of~\cite{8}). In Section~3 of
this paper we define the vague convergence of complex measures
on ${\mathbb R}^\nu$ with locally finite total variation, and
in Proposition~3.1 we impose this convergence for the
coordinates $G^{(N)}_{j,j'}$ of the sequence  spectral measures
$G^{(N)}$.

In Lemma 8.3 of \cite{6} the scalar-valued version of
Proposition~3.1 is formulated. In that result, the notion
of vague convergence also appears. But the definition of
the vague convergence in \cite{6} and in this paper is
slightly different. In~\cite{6} the vague convergence
of locally finite measures, while here the vague convergence
of complex measures on ${\mathbb R}^\nu$ with locally finite
total variation is defined. The difference between these two
definitions is greater than it may seem at the first sight.
Let me remark that while the restriction of a complex
measure on ${\mathbb R}^\nu$ with locally finite total
variation to the measurable subsets of a compact subset of
${\mathbb R}^\nu$ is a complex measure, there are such
complex measures on ${\mathbb R}^\nu$ with locally
finite total variation which are not complex measures
on the measurable subsets of ${\mathbb R}^\nu$. Hence we
are considering the vague convergence of different objects
in this paper and in~\cite{6}.

As I mentioned before, Proposition~3.1 is a multivariate
generalization of Lemma~8.3 in~\cite{6}. Its proof applies
the ideas of the proof of this lemma, but some additional
difficulties also had to be overcome.

Section~3 contains still another result. It is Lemma~3.2 which
may be useful when we apply Proposition~3.1. It states that if
all coordinates $G^{(N)}_{j,j'}$ of a sequence of spectral measures 
$G^{(N)}=(G^{(N)}_{j,j'})$, $1\le j,j'\le d$, $N=1,2,\dots$, converge 
vaguely to some complex measures $G^{(0)}_{j,j'}$ with finite total 
variation, then also $G^{(0)}=(G^{(0)}_{j,j'})$, $1\le j,j'\le d$, is a 
positive semidefinite matrix valued even measure on~$\mathbb R^\nu$.

This result is useful because it helps us to decide when the
limit matrix $G^{(0)}=(G^{(0)}_{j,j'})$, $1\le j,j'\le d$, is the
spectral measure of a (generalized) stationary Gaussian random 
field. (See the remark after the formulation of Lemma~3.2).

\medskip
In this paper together with~\cite{8} and also in the work~\cite{6}
I applied a version of the multiple Wiener--It\^o integrals
introduced by It\^o in his paper~\cite{5}. (It\^o called these
random integrals multiple Wiener integrals in his paper.) I applied,
instead of the random integral introduced by It\^o a version
of it introduced by Dobrushin in~\cite{3a}, and considered its
generalization when vector-valued stationary Gaussian random
fields are taken. At the end of the Introduction, I give
a short explanation why it was useful to work with this
version of the random integrals introduced by It\^o in~\cite{5}, 
and I also briefly mention another example where a
multidimensional time generalization of the It\^o integral was
introduced in order to study some problems.

It\^o considered in his paper~\cite{5} a measure space $(T,{\mathcal B},m)$,
and a set of (jointly) Gaussian random variables $\beta(E)$ indexed
by such sets $E\in{\mathcal B}$ for which $m(E)<\infty$, and the joint
distibution of these random variables is determined by the relations
$E\beta(E)=0$, and $E\beta(E)\beta(E')=m(E\cap E')$. He also imposed
the following continuity property for the measure~$m$. For any set
$E\in{\mathcal B}$ with $m(E)<\infty$ and $\varepsilon>0$ there
exists a finite decomposition $E=\sum_{i=1}^n E_i$ such that
$m(E_i)<\varepsilon$ for all $i=1,\dots,n$. It\^o defined the
$p$-fold Wiener integrals of square integrable functions (with
respect to the product measure $m^p$) with respect to the random
measure $\beta(E,\omega)$ for all $p=0,1,\dots$ In the definition
of the random measure $\beta(E,\omega)$ the parameter set consists
of those measurable sets $E$ for which $m(E)<\infty$. Then he gave
a useful representation of all square integrable random variables
measurable with respect to the $\sigma$-algebra generated by the
random variables $\beta(E)$ as a sum of multiple Wiener integrals
with different multiplicity.

To present such a representation It\^o proved some useful results
about the properties of multiple Wiener integrals. In particular,
he proved an identity that was later called It\^o's formula. He
showed with its help an important relation between multiple Wiener
integrals and Hermite polynomials.

In paper~\cite{3a} Dobrushin proved for the sake of investigation of
non-linear functionals of stationary Gaussian random fields such a
version of the random integrals in It\^o's paper~\cite{5}, where
he worked in the Euclidean space $({\mathbb R}^\nu,{\mathcal B})$,
and he replaced the random measure  $\beta(E,\omega)$ by the
random spectral measure of a stationary Gaussian random field.
I described these results in more detail in~\cite{6}, and this
paper together with~\cite{8} yield a generalization of these
results when vector-valued stationary Gaussian random fields
are considered. These works show that results similar to those
of the paper~\cite{3a} hold if we work with random spectral
measures instead of the random measure $\beta(E,\omega)$ applied
in~\cite{3a}. One may ask why it was useful to work with random
spectral measures instead of the application of the results
in~\cite{3a}.

I want to show that by working with random spectral measures
we could prove some Fourier analysis type results which were
useful in our investigation. Proposition~2.4 of this paper
is such a result. Here we calculate the shift transform $T_u$
of a random variable presented in the form of a multiple
Wiener--It\^o integral. The shifted random variable is presented
in a Fourier transform type form. Indeed, Proposition~2.4 says
that the shift transform $T_uY$ of a random variable $Y$ given
in formula~(\ref{2.6}) equals the multiple Wiener--It\^o
integral given in formula~(\ref{2.8}). This result together
with It\^o's formula which enables us very often to rewrite the
random variables we are working with as sums of multiple
Wiener--It\^o integrals help in the study of limit theorems.
The discussion at the beginning of Section~3 shows an example
of how to apply these results.

The application of multiple stochastic integrals turned out
to be useful also in the investigation of some other problems.
I mention only very briefly the contribution of
Wong--Zakai--Yor to the theory of multiple stochastic integrals
as an example for it. They discussed the following problem. 
The study of It\^o integrals is very closely related to the
study of martingales. For example, any continuous-time square
integrable martingale with continuous trajectories adapted to a
Wiener process has a canonical representation as an It\^o
integral with respect to the underlying Wiener process. The
above-mentioned mathematicians were looking for the
multidimensional time version of this result. Naturally, to
formulate this result first the multidimensional time Wiener
processes and martingales must be defined. But these objects
are defined in the literature. I would mention that in
particular, the definition of the multidimensional time
martingales demands special attention.

A multidimensional version of the result about the canonical
representation of square-integrable martingales can be proved.
But in that representation, not only the It\^o integrals
(integrals with multidimensional time which also have to be
defined) but also multiple It\^o integrals appear. The
precise formulation of this result would demand a very long
explanation, hence I omit it. The interested reader can find it
in paper~\cite{4a} and in its list of references. This work
also discusses with the help of this result a statistical
problem, where the main point of the solution is the
calculation of a Radon--Nikodym derivative.

\section{Wick polynomials and their relation to multipe
 Wie\-ner--It\^o integrals}

In the case of scalar-valued stationary Gaussian random  fields 
(i.e., if $d=1$) there is a so-called It\^o formula (see 
Theorem~4.3 in~\cite{6}) which shows an important relation 
between multiple Wiener--It\^o integrals and Hermite polynomials.
Here I present its multivariate version, where Wick polynomials 
take the role of the Hermite polynomials. Wick polynomials are the 
natural multi-dimensional generalizations of Hermite polynomials.
I shall also discuss an important consequence of the multivariate
version of the It\^o formula. This formula enables us to present 
a large class of random variables in the form of a sum of multiple 
Wiener--It\^o integrals. Besides, there is a useful formula 
for the calculation of the shift transforms of such random
variables which are given in the form of a sum of multiple 
Wiener--It\^o integrals. As we shall see, this formula is very
useful in the study of limit theorems for non-linear 
functionals of a vector-valued stationary Gaussian field. I shall
explain in the first part of the Appendix the relation between
the proof of It\^o's formula in the scalar and in the vector-valued
case. In that explanation I also write about the definition of
the Wick polynomials and their role in the proof.

\medskip
First I recall the definition of Wick polynomials and some 
results about their most important properties. Here I follow 
the discussion in Section~2 of~\cite{6}. 

Let $X_t$, $t\in T$, be a set of jointly Gaussian random 
variables indexed by a parameter set $T$, and such that 
$EX_t=0$ for all $t\in T$. We define the following real 
Hilbert spaces ${\mathcal H}_1$ and ${\mathcal H}$. A square 
integrable (real valued) random variable is in ${\mathcal H}$ 
if and only if it is measurable with respect to the 
$\sigma$-algebra ${\mathcal B}={\mathcal B}(X_t,\;t\in T)$, and 
the scalar product in ${\mathcal H}$ is defined as 
$\langle\xi,\eta\rangle=E\xi\eta$, $\xi,\,\eta\in{\mathcal H}$. 
The Hilbert space ${\mathcal H}_1\subset{\mathcal H}$ is the subspace 
of ${\mathcal H}$ generated by the finite linear combinations 
$\sum c_jX_{t_j}$, $t_j\in T$, with real coefficients. We
consider only such sets of Gaussian random variables 
$X_t$ for which ${\mathcal H}_1$ is separable. Otherwise 
$X_t$, $t\in T$, can be arbitrary, but the most interesting 
case for us is when $T={\mathbb Z}^\nu\times\{1,\dots,d\}$, 
and the original Gaussian random variables we are working 
with are the coordinates $X_j(p)$, $j\in\{1,\dots,d\}$, 
$p\in{\mathbb Z}^\nu$, of a vector-valued stationary Gaussian 
random field $X(p)=(X_1(p),\dots,X_d(p))$, $p\in{\mathbb Z}^\nu$.

To define the Wick polynomials and to get their most important 
properties we need the following result formulated in 
Theorem~2.1 of~\cite{6}.

\medskip\noindent
{\bf Theorem 2A.} {\it Let $Y_1,Y_2,\dots$ be an orthonormal basis
in the Hilbert space ${\mathcal H}_1$ 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 ${\mathcal H}$
defined above. (Here, and in the subsequent discussion $H_j(\cdot)$ 
denotes the $j$-th Hermite polynomial with leading coefficient~1.)}

\medskip
Let ${\mathcal H}_{\le n}\subset{\mathcal H}$, $n=1,2,\dots$, (with the 
previously introduced Hilbert space ${\mathcal H}$) denote the linear 
subspace of the Hilbert space ${\mathcal H}$ 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 
${\mathcal H}_0={\mathcal H}_{\le 0}$ consist of the constant functions, 
and let ${\mathcal H}_n={\mathcal H}_{\le n}\ominus{\mathcal H}_{\le n-1}$, 
$n=1,2,\dots$, where $\ominus$ denotes orthogonal completion. It is 
clear that the Hilbert space ${\mathcal H}_1$ given in this definition 
agrees with the previously defined Hilbert space ${\mathcal H}_1$. 
If $\xi_1,\dots,\xi_m\in{\mathcal H}_1$, and $P_n(x_1,\dots,x_m)$ is 
a polynomial of degree $n$, then
$P_n(\xi_1,\dots,\xi_m)\in{\mathcal H}_{\le n}$. Then Theorem~2A 
implies that
\begin{equation}
{\mathcal H}={\mathcal H}_0+{\mathcal H}_1+{\mathcal H}_2+\cdots,  
\label{2.1}
\end{equation}
where $+$ denotes direct sum. Now I present the definition of Wick
polynomials.

\medskip\noindent
{\bf Definition of Wick polynomials.} {\it Let $P(x_1,\dots,x_m)$ 
be a homogeneous polynomial of degree~$n$, and let a set of 
(jointly Gaussian) random variables $\xi_1,\dots,\xi_m\in{\mathcal H}_1$
be given. The Wick polynomial $\colon\!P(\xi_1,\dots,\xi_m)\colon$
determined by them is the orthogonal projection of the random variable 
$P(\xi_1,\dots,\xi_m)$ to the above-defined subspace ${\mathcal H}_n$ 
of the Hilbert space ${\mathcal H}$. The Wick polynomial of a
homogeneous polynomial of degree~$n$ will be called a Wick polynomial
of order~$n$.}

\medskip
In the sequel we shall use the notation 
$\colon\!P(\xi_1,\dots,\xi_m)\colon$ for the Wick polynomial 
corresponding to a homogeneous polynomial $P(x_1,\dots,x_m)$ 
with arguments $\xi_1,\dots,\xi_m$, $\xi_j\in{\mathcal H}_1$ for 
all $1\le j\le m$. It may happen that a random variable 
$\zeta$ can be expressed in two different forms as a 
homogeneous polynomial of some random variables from 
${\mathcal H}_1$, i.e., $\zeta=P_1(\xi_1,\dots,\xi_m)$, and 
$\zeta=P_2(\xi_1,\dots,\xi_m)$, and $P_1\neq P_2$. But 
in such a case
$$
\colon\!P_1(\xi_1,\dots,\xi_m)\colon=\colon\!P_2(\xi_1,\dots,\xi_m)\colon,
$$ 
i.e., the value of a Wick polynomial
$\colon\!P(\xi_1,\dots,\xi_m)\colon$ does not depend on the
representation of the random variable $P(\xi_1,\dots,\xi_m)$.

\medskip
It is clear that Wick polynomials of different degree are 
orthogonal. Given some $\xi_1,\dots,\xi_m\in{\mathcal H}_1$ 
define the subspaces
${\mathcal H}_{\le n}(\xi_1,\dots,\xi_m)\subset{\mathcal H}_{\le n}$, 
$n=1,2,\dots$, as the set of all polynomials of the random 
variables $\xi_1,\dots,\xi_m$ with degree less than or equal 
to~$n$. Let
${\mathcal H}_{\le0}(\xi_1,\dots,\xi_m)
={\mathcal H}_{0}(\xi_1,\dots,\xi_m)={\mathcal H}_0$,
and 
${\mathcal H}_n(\xi_1,\dots,\xi_m)
={\mathcal H}_{\le n}(\xi_1,\dots,\xi_m)\ominus
{\mathcal H}_{\le n-1}(\xi_1,\dots,\xi_m)$. With the help of this notation I
formulate the following result given in~Proposition~2.2 of~\cite{6}.

\medskip\noindent
{\bf Theorem 2B.} {\it Let $P(x_1,\dots,x_m)$ be a homogeneous 
polynomial of degree~$n$. Then $\colon\!P(\xi_1,\dots,\xi_m)\colon$ 
equals the orthogonal projection of $P(\xi_1,\dots,\xi_m)$ to   
${\mathcal H}_n(\xi_1,\dots,\xi_m)$.}

\medskip 
This result has the following important consequences formulated in
Corollaries~2.3 and~2.4 in~\cite{6}.

\medskip\noindent
{\bf Corollary 2C.} {\it Let $\xi_1,\dots,\xi_m$ be an orthonormal
system in ${\mathcal H}_1$, 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
$$
\colon\!P(\xi_1,\dots,\xi_m)\colon=\sum
c_{j_1,\dots,j_m}H_{j_1}(\xi_1)\cdots H_{j_m}(\xi_m).
$$
In particular,
$$
\colon\!\xi^n\colon=H_n(\xi) \quad \textrm{if } \xi\in{\mathcal H}_1, 
\textrm{ and } E\xi^2=1.
$$
}

\medskip\noindent
{\bf Corollary 2D.} {\it Let $\xi_1,\xi_2,\dots$ be an orthonormal
basis in ${\mathcal H}_1$. 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 ${\mathcal H}_n$.}

\medskip
In the proof of the It\^o formula for scalar-valued stationary 
random fields we needed, besides the diagram formula, the 
following important recursive formula for Hermite polynomials 
which is contained for example in Lemma~5.2 of~\cite{6}.
\begin{equation}
H_n(x)=xH_{n-1}(x)-(n-1)H_{n-2}(x) \quad\textrm{for \ }n=1,2,\dots, 
\label{2.2}
\end{equation}
with the notation $H_{-1}(x)\equiv0$ in the case $n=1$.

In the next result I formulate a multivariate version of 
this formula for Wick polynomials. 

\medskip\noindent
{\bf Proposition 2.1.} {\it Let $U_1,\dots,U_{n+1}$, $n\ge1$, be 
elements in ${{\mathcal H}_1}$. Then
\begin{eqnarray}
&&\colon\! U_1\cdots U_n\colon U_{n+1} \label{2.3} \\
&&\qquad=\colon\! U_1\cdots U_n U_{n+1}\colon
+\sum_{s=1}^n \colon\! U_1\cdots U_{s-1}U_{s+1}\cdots U_n\colon
 EU_sU_{n+1}.  \nonumber 
\end{eqnarray}
In the special case $n=1$ this formula is meant as 
$U_1U_2=\colon\!U_1U_2\colon+EU_1U_2$.}

\medskip\noindent
{\it Proof of Proposition 2.1.}\/ Formula (\ref{2.3}) clearly holds if 
all random variables $U_j$, $1\le j\le n+1$ agree, and $EU_1^2=1$, 
since in this case the left-hand side of (\ref{2.3}) equals $U_1H_n(U_1)$, 
while its right-hand side equals $H_{n+1}(U_1)+nH_{n-1}(U_1)$ by 
Corollary~2C, and these two expressions are equal by formula (\ref{2.2}). 
A somewhat more complicated, but similar argument shows that this 
formula also holds if the sequence $U_1,\dots,U_n$ consists of some 
independent random variables $V_1\dots,V_k$ with standard normal 
distribution, the random variable $V_p$ is contained in the sequence 
$U_1,$\dots, $U_n$ with multiplicity~$l_p$, $1\le p\le k$, and 
finally $U_{n+1}$ is either one of these random variables $V_p$, 
$1\le p\le k$, or it is a random variable $V_{k+1}$ with standard
normal distribution which is independent of all of them. 

Indeed, if $U_{n+1}=V_p$ with some $1\le p\le k$, then the left-hand 
side of~(\ref{2.3}) equals 
$$
H_{l_1}(V_1)\cdots H_{l_k}(V_k)V_p,
$$ 
while the right-hand side equals 
\begin{eqnarray*}
&&H_{l_1}(V_1)\cdots H_{l_{p-1}}(V_{p-1})H_{l_p+1}(V_p)H_{l_{p+1}}(V_{p+1})
\cdots H_{l_k}(V_k) \\
&&\qquad +l_p \, 
H_{l_1}(V_1)\cdots H_{l_{p-1}}(V_{p-1})H_{l_p-1}(V_p)H_{l_{p+1}}(V_{p+1})
\cdots H_{l_k}(V_k)
\end{eqnarray*}
by Corollary~2C. A comparison of these expressions together with 
relation~(\ref{2.2}) imply that identity~(\ref{2.3}) holds 
in this case. If $U_{n+1}=V_{k+1}$, then the left-hand side 
of~(\ref{2.3}) equals
$$
H_{l_1}(V_1)\cdots H_{l_k}(V_k)V_{k+1},
$$ 
and the right-hand side also equals $H_{l_1}(V_1)\cdots H_{l_k}(V_k)V_{k+1}$. 
Hence formula~(\ref{2.3}) holds in this case, too.

In the general case we can choose some independent Gaussian 
random variables $Z_1,\dots,Z_m$ in ${\mathcal H}_1$ with 
variance~1 in such a way that our random variables 
$U_1,\dots,U_{n+1}$ can be expressed as their linear 
combination, i.e., $U_p=\sum_{l=1}^m c_{p,l}Z_l$ with some 
coefficients $c_{l,m}$. We have already seen that formula~(\ref{2.3}) 
is valid in the special case when all random variables $U_p$ 
equal one of the random variables $Z_j$, i.e., if $U_p=Z_{j(p)}$ 
with some $1\le j(p)\le m$ for all $1\le p\le n+1$. Since the 
expressions of both sides of~(\ref{2.3}) are multi-linear functionals 
on the $n$-fold direct product 
${\mathcal H}_1\times\cdots\times{\mathcal H}_1$, this implies that 
formula~(\ref{2.3}) also holds for the random 
variables~$U_1,\dots,U_{n+1}$. Proposition~2.1 is proved. \qed 

\medskip
We can prove the multivariate version of It\^o's 
formula with the help of Proposition~2.1 and the diagram 
formula for multiple Wiener--It\^o integrals for 
vector-valued stationary Gaussian random fields formulated in
Section~6 of~\cite{8}.

Before its formulation, I make a remark about the notation 
in this section.

In the formulation of It\^o's formula the notation 
${\mathcal K}_{1,j}$ appears. This notion was introduced in 
Lemma~3.2 of~\cite{8}. It is a real Hilbert space, and it 
contains those functions $u\in{\mathcal K}_{1,j}$ for which we 
defined the random integral $\int u(x)Z_{G,j}(\,dx)$ with 
respect to the $j$-th coordinate $Z_{G,j}$ of the random 
spectral measure $Z_G=(Z_{G,1},\dots,Z_{G,d})$, and the value 
of this integral is a real valued random variable. In Section~4 
of~\cite{8} this notion is defined also in the case of 
generalized random spectral measure. Later, at the beginning
of Section~5 of~\cite{8} its multidimensional
generalization, the real Hilbert space
 ${\mathcal K}_{n,j_1,\dots,j_n}
={\mathcal K}_{n,j_1,\dots,j_n}(G_{j_1,j_1}.\dots,G_{j_n,j_n})$ is defined 
for all $n=1,2,\dots$. It consists of those functions 
$f(x_1,\dots,x_n)$ for which the $n$-fold (real valued) 
Wiener--It\^o integral 
$$
I_n(f|j_1,\dots,j_n)=\int f(x_1,\dots,x_n)
Z_{G,{j_1}}(\,dx_1)\dots Z_{G,{j_n}}(d\,x_n)
$$
is defined. The Hilbert space ${\mathcal K}_{1,j}$ is a special
case of these Hilbert spaces with $n=1$. The Hilbert spaces 
${\mathcal K}_{n,j_1,\dots,j_n}$ will appear also in this work. At
some points I shall also work with the class of simple functions
$\hat{\mathcal K}_{n,j_1,\dots,j_n}\subset{\mathcal K}_{n,j_1,\dots,j_n}$ 
defined also in Section~5 of~\cite{8}. The multiple Wiener--It\^o
integrals were first defined for simple functions which are adapted
to some regular system (defined also in Section~5 of~\cite{8}),
and the multiple Wiener--It\^o integrals were defined in the general
case by means of a good approximation of the functions
$f\in{\mathcal K}_{n,j_1,\dots,j_n}$ by simple functions
$f\in\hat{\mathcal K}_{n,j_1,\dots,j_n}$.

\medskip\noindent
{\bf Theorem 2.2. Multivariate version of It\^o's formula.} 
{\it Let us have some vector-valued stationary Gaussian  
random field with a vector-valued random spectral measure
$Z_G=(Z_{G,1},\dots,Z_{G,d})$. Let us consider some functions
$\varphi_p\in {\mathcal K}_{1,j_p}$, $1\le p\le n$,
$1\le j_p\le d$, and define with their help the random variables
$U_p=\int\varphi_p(x)Z_{G,{j_p}}(\,dx)\in{\mathcal H}_1$, $1\le p\le n$. 
The identity
\begin{eqnarray}
&&\colon\!U_1\cdots U_n\colon  \label{2.4} \\
&&\qquad=\int \varphi_1(x_1)\varphi_2(x_2)\cdots \varphi_n(x_n)
Z_{G,{j_1}}(\,dx_1) Z_{G,{j_2}}(\,dx_2)\cdots Z_{G,{j_n}}(\,dx_n) \nonumber 
\end{eqnarray}
holds.}

\medskip\noindent
{\it Proof of Theorem 2.2.}\/ Relation~(\ref{2.4}) clearly 
holds for $n=1$. We prove by induction that it holds for 
$n+1$ if it holds for $k\le n$. In the proof we apply the 
Corollary of Theorem~6.1 from~\cite{8}, (i.e., the corollary
of the diagram formula from that paper) with the choice
\begin{eqnarray*}
h_1(x_1,\dots,x_n)&=&\varphi_1(x_1)\cdots\varphi_n(x_n) \\
h_2(x)&=&\varphi_{n+1}(x),
\end{eqnarray*}
and the random spectral measure $Z_{G,{j'_1}}$ is chosen as 
$Z_{G,j'_1}=Z_{G,j_{n+1}}$, where 
$Z_{G,{j_{n+1}}}$ is the random spectral measure appearing in the 
definition of $U_{n+1}$. We can write with this choice the identity  
\begin{eqnarray}
&&   \!\!\!\!\!\!\!\!\!\! \int \varphi_1(x_1)\cdots\varphi_n(x_n)
Z_{G,{j_1}}(\,dx_1)\cdots Z_{G,{j_n}}(\,dx_n)
\int \varphi_{n+1}(x) Z_{G,{j_{n+1} }}(\,dx)  \label{2.5} \\
&&=
\int \varphi_1(x_1)\cdots\varphi_n(x_n)\varphi_{n+1}(x_{n+1})
Z_{G,{j_1}}(\,dx_1)\cdots Z_{G,{j_{n+1}}}(\,dx_{n+1}) \nonumber \\
&&\qquad+\sum_{p=1}^n EU_pU_{n+1}  
\int \varphi_1(x_1)\cdots\varphi_{p-1}(x_{p-1})\varphi_{p+1}(x_p)
\cdots \varphi_n(x_{n-1})  \nonumber \\
&&\qquad\qquad Z_{G,{j_1}}(\,dx_1)\cdots 
Z_{G,{j_{p-1} }}(\,dx_{p-1}) Z_{G,{j_{p+1} }}(\,dx_p) \cdots
Z_{G,{j_n}}(\,dx_{n-1}), \nonumber
\end{eqnarray}
since formula~(6.19) in \cite{8} gives this identity with 
our choice of $h_1$ and $h_2$. To see this observe that
with these functions $h_1$ and $h_2$ the function 
$h_{\gamma_p}$ in the formulation of the corollary of Theorem~6.1 
in~\cite{8} equals for $p\neq0$
\begin{eqnarray*}
h_{\gamma_p}(x_1,\dots,x_{n-1})&=&
\varphi_1(x_1)\cdots\varphi_{p-1}(x_{p-1})\varphi_{p+1}(x_p)\cdots
\varphi_n(x_{n-1})\\ 
&&\qquad \int \varphi_p(x_n)\overline{\varphi_{n+1}(x_n)}
G_{j_p,j_{n+1}}(\,dx_n)\\
&=&\varphi_1(x_1)\cdots\varphi_{p-1}(x_{p-1})\varphi_{p+1}(x_p)
\cdots\varphi_n(x_{n-1}) EU_pU_{n+1}
\end{eqnarray*}
since $U_{n+1}=\overline{U_{n+1}}$, and by formula~(3.7) 
in~\cite{8} which expresses the scalar product of
two one-fold random integrals 
\begin{eqnarray*}
EU_pU_{n+1}&=&EU_p\overline{U_{n+1}}
=E\left(\int \varphi_p(x)Z_{G,j_p}(\,dx)\overline{\int \varphi_{n+1}(x)
Z_{G,{j_{n+1}}}(\,dx)}\right) \\
&=&\int \varphi_p(x_n)\overline{\varphi_{n+1}(x_n)}G_{j_p,j_{n+1}}(\,dx_n), 
\end{eqnarray*}
and for $p=0$
$$
h_{\gamma_0}(x_1,\dots,x_{n+1})=
\varphi_1(x_1)\cdots\varphi_n(x_n) \varphi_{n+1}(x_{n+1}).
$$
Corollary of Theorem~6.1 in~\cite{8} with the above form of
the kernel functions $h_{\gamma_p}$, $0\le p\le n$, in it imply
formula~(\ref{2.5}). Formula~(\ref{2.5}) together with our 
induction hypothesis imply that
\begin{eqnarray*}
&&\int \varphi_1(x_1)\cdots\varphi_n(x_n)\varphi_{n+1}(x_{n+1})
Z_{G,{j_1}}(\,dx_1)\cdots Z_{G,{j_{n+1}}}(\,dx_{n+1}) \\
&&\qquad =\colon\! U_1\cdots U_n\colon U_{n+1}-
\sum_{p=1}^n \colon\! U_1\cdots U_{p-1}U_{p+1}\cdots U_n\colon EU_p U_{n+1}.
\end{eqnarray*}
In the case $n=1$ this formula means that
$$
\int \varphi_1(x_1)\varphi_2(x_2)Z_{G,{j_1}}(\,dx_1)Z_{G,{j_2}}(\,dx_2) =
U_1U_2-EU_1U_2.
$$ 
By comparing the last formula with~(\ref{2.3}) we get that
the statement of Theorem~2.2 holds also for $n+1$. Theorem~2.2 is
proved. \qed

\medskip
In Theorem~2.2 we rewrote some Wick polynomials of special form
as multiple Wiener--It\^o integrals. This enables us to express 
a sum of such Wick polynomials as the sum of multiple Wiener--It\^o
integrals. This implies that all Wick polynomials of random variables
from some ${\mathcal H}_{1,j}$, $1\le j\le d$, can be written in
the form of a sum of multiple Wiener--It\^o integrals. In the next
simple corollary of Theorem~2.2  I describe this result in a
more explicit form.

To formulate this result let us introduce the following notation. Let
us fix some numbers $n\ge1$ (the order of the homogeneous polynomial
we are considering), $m\ge1$ and some functions 
$\varphi_{j,k}(x)\in {\mathcal K}_{1,j}$, $1\le j\le d$, $1\le k\le m$,
and define the random variables
$$
\xi_{j,k}=\int \varphi_{j,k}(x) Z_{G,j}(\,dx), \quad 1\le j\le d,\; 
1\le k\le m.
$$
Then $\xi_{j,k}\in{\mathcal H}_{1,j}$. (We defined the real Hilbert space
${\mathcal H}_{1,j}$  in the formulation of Lemma~3.2 of~\cite{8}. This 
Lemma~3.2 stated that the elements of ${\mathcal H}_{1,j}$
can be given in the form of the above integral.)  

In the next corollary, we consider homogeneous polynomials 
of these random variables $\xi_{j,k}$, and express the Wick 
polynomials corresponding to them in the form of a sum 
of multiple Wiener--It\^o integrals.

\medskip\noindent
{\bf Corollary of Theorem 2.2.} {\it Let us consider a homogeneous
polynomial
\begin{eqnarray*}
&&P(x_{j_s,k_s},\; 1\le j_s\le d,\; 1\le k_s\le m 
\textrm{ for all } 1\le s\le n)\\
&&\qquad =\sum_{\substack{1\le j_s\le d \textrm{ for all } 1\le s\le n \\
 1\le k_s\le m \textrm{ for all } 1\le s\le n}}
a_{j_1,k_1,\dots j_n,k_n}x_{j_1,k_1}x_{j_2,k_2}\cdots x_{j_n,k_n}
\end{eqnarray*}
of order $n$ of the variables $x_{j_s,k_s}$ with
indices $1\le j_s\le d$ and $1\le k_s\le m$ for all $1\le s\le n$
and real coefficients $a_{j_1,k_1,\dots,j_n,k_n}$.

If we replace the variables $x_{j_s,k_s}$ with the random variables
$$
\xi_{j_s,k_s}=\int \varphi_{j_s,k_s}(x) Z_{G,j_s}(\,dx)
$$ 
 in this polynomial (we choose a function
$\varphi_{j,k}\in{\mathcal K}_{1,j}$ in the definition of $\xi_{j,k}$), 
then we get the following homogeneous polynomial of some 
jointly Gaussian random variables.
\begin{eqnarray*}
&&P(\xi_{j_s,k_s},\; 1\le j_s\le d,\; 1\le k_s\le m \textrm{ for all } 
1\le s\le n)\\
&&\qquad =\sum_{\substack{1\le j_s\le d \textrm{ for all } 1\le s\le n \\
 1\le k_s\le m \textrm{ for all } 1\le s\le n}}
a_{j_1,k_1,\dots j_n,k_n}\xi_{j_1,k_1}\xi_{j_2,k_2}\cdots \xi_{j_n,k_n}.
\end{eqnarray*}
With the help of this expression we can define the Wick polynomial
$$
\colon\!P(\xi_{j_s,k_s},\; 1\le j_s\le d,\; 1\le k_s\le m 
\textrm{ for all } 1\le s\le n)\colon .
$$ 
This Wick polynomial can be expressed as a sum of multiple
Wiener--It\^o integrals in the following way.

Let us consider for all sequences of indices $\{(j_s,k_s), 
\colon\;1\le s\le n\}$ with $1\le j_s\le d$, $1\le k_s\le d$ for all 
$1\le s\le n$ the function 
$$
f_{j_1,k_1,\dots,j_n,k_n}(x_1,\dots,x_n)=
\varphi_{j_1,k_1}(x_1)\cdots\varphi_{j_n,k_n}(x_n)
\in{\mathcal K}_{n,j_1,\dots,j_n}
$$
and the multiple Wiener--It\^o integral
\begin{eqnarray*}
&&I_n(f_{j_1,k_1,\dots,j_n,k_n}|j_1,\dots,j_n) \\
&&\qquad=\int f_{j_1,k_1,\dots,j_n,k_n}(x_1,\dots,x_n)
Z_{G,{j_1}}(\,dx_1) \dots Z_{G,{j_n}}(\,dx_n).
\end{eqnarray*}
The identity
\begin{eqnarray*}
&&\colon\!P(\xi_{j_s,k_s},\; 1\le j_s\le d,\; 1\le k_s\le m 
\textrm{ for all } 1\le s\le n)\colon \\
&&\qquad =\sum_{\substack{1\le j_s\le d \textrm{ for all } 1\le s\le n \\
 1\le k_s\le m \textrm{ for all } 1\le s\le n}}
a_{j_1,k_1,\dots j_n,k_n} I_n(f_{j_1,k_1,\dots,j_n,k_n}|j_1,\dots,j_n)
\end{eqnarray*}
holds.}

\medskip\noindent
{\it Remark.}\/ Theorem~4.7 of~\cite{6} contains a version of this
result for scalar-valued stationary Gaussian random fields.

\medskip\noindent
{\it Proof of the Corollary of Theorem 2.2.}\/ By Theorem~2.2 
we have
$$
a_{j_1,k_1,\dots j_n,k_n}\colon\!\xi_{j_1,k_1}\xi_{j_2,k_2}\cdots \xi_{j_n,k_n}\colon=
a_{j_1,k_1,\dots j_n,k_n} I_n(f_{j_1,k_1,\dots,j_n,k_n}|j_1,\dots,j_n)
$$
for all sequences of indices $(j_s,k_s)$, $1\le s\le n$. By summing 
up these inequalities for all sequences of indices we get the proof 
of the corollary.  \qed

\medskip
With the help of the above corollary, we prove the following result.

\medskip\noindent
{\bf Proposition 2.3.} {\it For all $n\ge 1$ and functions 
$f\in{\mathcal K}_{n,j_1,\dots,j_n}$ with some indices $1\le j_s\le d$, 
$1\le s\le n$, $I_n(f|j_1,\dots,j_n)\in{\mathcal H}_n$ for the 
$n$-fold Wiener--It\^o integral $I_n(f|j_1,\dots,j_n)$. Besides, 
the set of all  sums of $n$-fold Wiener--It\^o integrals i.e., 
the set of all sums of the form
$$
\sum_{1\le j_s\le d \textrm{ for all }1\le s\le n} 
I_n(f_{j_1,\dots,j_n}|j_1,\dots,j_n),
$$
where $f_{j_1,\dots,j_n}\in{\mathcal K}_{n,j_1,\dots,j_n}$
constitute an everywhere dense linear subspace of ${\mathcal H}_n$.}

\medskip\noindent
{\it Proof of Proposition 2.3.}\/ We shall prove 
Proposition~2.3 by induction with respect to~$n$. 
By Lemma~3.2 of~\cite{8} Proposition 2.3 holds for $n=1$. 
Indeed, by this result every random variable of the
form $\xi=\sum_{j=1}^d\xi_j$ with some 
$\xi_j\in{\mathcal H}_{1,j}$ can be written as the sum
of one-fold Wiener--It\^o integrals, and the random
variables of this form constitute an everywhere dense
linear subspace of ${\mathcal H}_1$.

If the statements of Proposition~2.3 hold for all $m<n$, 
then we can say for one part that 
$I_n(f|j_1,\dots,j_n)\in {\mathcal H}_{\le n}$, because this 
relation holds if $f$ is a simple function, i.e., if 
$f\in\hat{\mathcal K}_{n,j_1,\dots,j_n}$ with the space 
$\hat{\mathcal K}_{n,j_1,\dots,j_n}$ defined in Section~5 
of~\cite{8}, and  since $\hat{\mathcal K}_{n,j_1,\dots,j_n}$ 
is dense in ${\mathcal K}_{n,j_1,\dots,j_n}$, and we defined 
the multiple Wiener--It\^o integral by the extension of a
bounded operator in the general case, the above property remains 
valid for general functions $f\in{\mathcal K}_{n,j_1,\dots,j_n}$. 
Moreover, we know that $I_n(f|j_1,\dots,j_n)$ is 
orthogonal to all multiple Wiener--It\^o integrals of the form 
$I_m(h|j'_1,\dots,j'_m)$ with $m<n$ because of 
relation~(5.5) in~\cite{8}. Then $I_n(f|j_1,\dots,j_n)$ 
is also orthogonal to any linear combination of such 
integrals. But these linear combinations constitute an 
everywhere dense set in ${\mathcal H}_m$ by our inductive 
hypothesis. Hence $I_n(f|j_1,\dots,j_n)$ is orthogonal 
to the whole space ${\mathcal H}_m$ for all $0\le m\le n-1$, 
and this implies that it is contained in the Hilbert 
subspace ${\mathcal H}_n$ (and not only in ${\mathcal H}_{\le n}$).
It follows from the corollary of Theorem~2.2 that the sums
of multiple Wiener--It\^o integrals considered in
Proposition~2.3 are dense in ${\mathcal H}_n$, 
and they constitute a linear subspace. Indeed, this 
corollary implies that a large class of Wick polynomials of
order $n$ can be expressed as a sum of such integrals, and 
the class of these Wick polynomials of order~$n$ is dense in
${\mathcal H}_n$. Proposition~2.3 is proved.  \qed

\medskip\noindent
{\it Remark.} In Proposition 2.3 we expressed a dense subset of 
${\mathcal H}_n$ as a sum of $n$-fold Wiener--It\^o integrals, but 
we did not express all elements of ${\mathcal H}_n$ in such a form. 
But even this weaker result suffices for our purposes.

In the case of scalar-valued stationary random fields, we have
a stronger result. In that case, we can express all elements of 
${\mathcal H}_n$ as an $n$-fold Wiener--It\^o integral, and actually, 
we can say somewhat more. There is a so-called Fock space 
representation of all elements $h\in{\mathcal H}$, which represents 
the elements $h\in {\mathcal H}$ in the form of a sum of multiple 
Wiener--It\^o integrals of different multiplicity. (See Theorem~4.2
of~\cite{6} together with the definition of Fock spaces on page~28
of~\cite{6}.) Moreover, this result has some useful consequences
about the properties of this representation.

We cannot prove a similar result in the vector-valued case. This
difference appears because of the following reason. If a sequence 
of random variables $h_N\in{\mathcal H}_n$, $N=0,1,2,\dots$, has 
the property that $h_N\to h_0$ with some $h_0\in{\mathcal H}_n$ in 
the norm of ${\mathcal H}_n$ as $N\to\infty$ in the scalar-valued 
case, then these random variables  $h_N$ can be expressed as $n$-fold 
Wiener--It\^o integrals of such functions $k_N\in{\mathcal K}_n$ 
for which $k_N\to k_0$ in the norm of ${\mathcal K}_n$. On the
other hands, in the case of vector-valued models we do not have a 
similar result. 

\medskip
Next, we consider a vector-valued stationary Gaussian random 
field 
$$
X(p)=(X_1(p),\dots,X_d(p)), \quad p\in{\mathbb Z}^\nu,
$$
whose elements can be written in the form 
$X_j(p)=\int e^{i(p,x)}Z_{G,j}(\,dx)$ by means of the 
random spectral measure $Z_G=(Z_{G,1},\dots,Z_{G,d})$
of this random field for all $p\in{\mathbb Z}^\nu$ 
and $1\le j\le d$. Let us consider a random 
variable $Y\in{\mathcal H}_n$ which can be represented 
as the $n$-fold Wiener--It\^o integral of some
function $h\in{\mathcal K}_{n,j_1,\dots,j_n}$, i.e.,
\begin{equation}
Y=\int h(x_1,\dots,x_n)Z_{G,{j_1}}(\,dx_1)\dots Z_{G,{j_n}}(\,dx_n). 
\label{2.6}
\end{equation}
I shall express the shift transforms $T_uY$, $u\in{\mathbb Z}^\nu$, 
of $Y$ given in formula~(\ref{2.6}) by an expression which 
shows some similarity to the Fourier transform of a function.  

To do this first I recall the definition of the shift transforms
$T_u$, $u\in{\mathbb Z}^\nu$, in a stationary random field 
$X(p)=(X_1(p),\dots,X_d(p))$, $p\in{\mathbb Z}^\nu$.

Given some element $X_j(m)$, $m\in{\mathbb Z}^\nu$, $1\le j\le d$, 
of the random field, and $u\in{\mathbb Z}^\nu$, we define 
the shift transform of $X_j(m)$ by $T_u$ as $T_uX_j(m)=X_j(u+m)$. 
More generally, given any measurable function 
$h(X_j(m), m\in{\mathbb Z}^\nu, 1\le j\le d)$,
we define the shift transform of the random variable
$Y=h(X_j(m),\, m\in{\mathbb Z}^\nu,\, 1\le j\le d)$, by the 
formula $T_uY=h(X_j(m+u),\, m\in{\mathbb Z}^\nu,\, 1\le j\le d)$.
This transformation was discussed in the scalar-valued case 
in~\cite{6}. It can be seen, (similarly to the argument in that work)
that the definition of this transformation is meaningful, (i.e., 
the value of $T_uY$ does not depend on the choice of the 
function~$h$ for which
$Y=h(X_j(m),\, m\in{\mathbb Z}^\nu,\, 1\le j\le d)$), and we 
have defined in such a way unitary (linear) transformations $T_u$,
$u\in{\mathbb Z}^\nu$, on ${\mathcal H}$ for which $T_uT_v=T_{u+v}$.

In Lemma~3.2 of~\cite{8} I have shown that each random variable 
$U_j\in{\mathcal H}_{1,j}$ can be written in the form 
$U_j=\int h(x)Z_{G,j}(\,dx)$ with some function 
$h(x)\in{\mathcal K}_{1,j}$. On the other hand, I claim that for
all $u\in{\mathbb Z}^\nu$
\begin{equation}
T_u U_j=\int e^{i(u,x)}h(x)Z_{G,j}(\,dx) \quad \textrm{if} \quad
U_j=\int h(x)Z_{G,j}(\,dx) \label{2.7}
\end{equation}
with some $h\in {\mathcal K}_{1,j}$. Indeed, relation~(\ref{2.7}) 
clearly holds if $h(x)=e^{i(p,x)}$ with some $p\in{\mathbb Z}^\nu$, 
since in this case $U_j=X_j(p)$ and  $T_uU_j=X_j(p+u)$. But 
this implies that relation (\ref{2.7}) holds for all finite 
trigonometrical polynomials of the form 
$h(x)=\sum c_k e^{i(p_k,x)}$, and for the closure of these 
functions with respect to the $L_2$ norm determined by
the measure $G_{j,j}$, i.e., for all $h\in{\mathcal K}_{1,j}$.

In Proposition~2.4 I present a similar formula about the shift 
transforms of a random variable~$Y$ given by formula~(\ref{2.6}). 
This result is useful in the study of limit theorems related to 
non-linear functionals of a stationary Gaussian field.

\medskip\noindent
{\bf Proposition 2.4 about the representation of shift 
transformations.} {\it Let a vector-valued stationary Gaussian 
random field 
$$
X(p)=(X_1(p),\dots,X_d(p)), \quad p\in{\mathbb Z}^\nu,
$$ 
be given with a vector-valued random spectral measure 
$Z_G=(Z_{G,1},\dots,Z_{G,d})$ such that  
$X_j(p)=\int e^{i(p,x)}Z_{G,j}(\,dx)$ for all $p\in{\mathbb Z}^\nu$ 
and $1\le j\le d$. Let $Y\in{\mathcal H}_n$ be  the random variable 
defined in formula~(\ref{2.6}) with the help of this vector-valued 
random spectral measure~$Z_G$ and some function 
$h\in{\mathcal K}_{n,j_1,\dots,j_n}$. Then
\begin{equation}
T_uY=\int e^{i(u,x_1+\cdots+x_n)}
 h(x_1,\dots,x_n)Z_{G,{j_1}}(\,dx_1)\dots Z_{G,{j_n}}(\,dx_n) 
\label{2.8}
\end{equation}
for all $u\in{\mathbb Z}^\nu$.
}

\medskip\noindent
{\it Proof of Proposition 2.4.}\/ Formula (\ref{2.8}) holds in the 
special case if $n=1$, and $h(x)\in {\mathcal K}_{1,j}$, since in 
this case $Y=\int h(x)Z_{G,j}(\,dx)$, and 
$$
T_uY=\int e^{i(u,x)} h(x)Z_{G,j}(\,dx)
$$ 
by formula~(\ref{2.7}).

I claim that formula~(\ref{2.8}) also holds in the case when the 
random variable $Y$ is given by formula~(\ref{2.6}) with a kernel 
function of the form
$h(x_1,\dots,x_n)=\varphi_1(x_1)\cdots\varphi_n(x_n)$ defined 
with the help of some functions $\varphi_s(x)\in{\mathcal K}_{1,j_s}$, 
$1\le s\le n$. Indeed, in this case $Y=\colon\!U_1\cdots U_n\colon$
with $U_s=\int\varphi_s(x)Z_{G,{j_s}}(\,dx)$, $1\le s\le n$,
because of Theorem~2.2. On the other hand, I claim that 
$$
T_u\colon\!U_1\cdots U_n\colon
=\colon\! (T_uU_1)\cdots (T_uU_n)\colon.
$$ 
To see this let us observe that by Theorem~2B 
$\colon\!U_1\cdots U_n\colon$ is the orthogonal projection of 
the product $U_1\cdots U_n$ to the Hilbert subspace 
${\mathcal H}_n(U_1,\dots,U_n)$. 
Similarly, $\colon\! (T_uU_1)\cdots (T_uU_n)\colon$ is the orthogonal 
projection of the vector $(T_uU_1)\cdots (T_uU_n)$ to the Hilbert 
subspace ${\mathcal H}_n(T_uU_1,\dots,T_uU_n)$. 
Since the vectors $(U_1,\dots,U_n)$ and $(T_u U_1,\dots, T_uU_n)$ 
have the same distribution, and the Wick polynomial corresponding 
to their product can be calculated in the same way this implies 
that if $\colon\!U_1\cdots U_n\colon=g(U_1,\dots,U_n)$ with some 
function $g$, then 
$$
\colon\! (T_uU_1)\cdots (T_uU_n)\colon=g(T_uU_1,\dots,T_uU_n)
$$ 
with the same function~$g$. (In the present case $g(x_1,\dots,x_n)$ 
is a polynomial of order~$n$.) On the other hand,
$T_u\colon\!U_1\cdots U_n\colon=T_ug(U_1,\dots,U_n)=g(T_uU_1,\dots,T_uU_n)$ 
in this case. The above argument implies the desired identity.

Thus we can state that if $Y$ is defined by formula~(\ref{2.6}) 
with a function
$$
h(x_1,\dots,x_n)=\varphi_1(x_1)\cdots\varphi_n(x_n)
$$
 with the above properties, then
\begin{eqnarray*}
T_uY&=&\colon\! (T_uU_1)\cdots (T_uU_n)\colon \\
&=&\int e^{i(u,x_1+\cdots+x_n)}
 h(x_1,\dots,x_n)Z_{G,{j_1}}(\,dx_1)\dots Z_{G,{j_n}}(\,dx_n) 
\end{eqnarray*}
because of Theorem~2.2 and the relation
$T_uU_s=\int e^{i(u,x)}\varphi_s(x)Z_{G,{j_s}}(\,dx)$ for all 
indices $1\le s\le n$.

From the result in the previous case follows that relation~(\ref{2.8})
also holds if $Y$ is defined by~(\ref{2.6}) with a function 
$h(x_1,\dots,x_n)$ of the form of a finite sum
$$
h(x_1,\dots,x_n)=\sum_k 
\varphi_{1,k}(x_1)\varphi_{2,k}(x_2)\cdots\varphi_{n,k}(x_n)
$$
with $\varphi_{s,k}\in{\mathcal K}_{1,j_s}$, $1\le s\le n$.

Since functions of the above form are dense in 
${\mathcal K}_{n,j_1,\dots,j_n}$, $T_u$ is a unitary operator, and 
both (linear) transformations 
$$
h(x_1,\dots,x_n)\to e^{i(u,x_1+\cdots+x_n)}h(x_1,\dots,x_n)
$$ 
and $h\to I_n(h|j_1,\dots,j_n)$ from the space
${\mathcal K}_{n,j_1,\dots,j_n}$ to the spaces ${\mathcal K}_{n,j_1,\dots,j_n}$ 
and ${\mathcal H}_n$ are of bounded norms, it is not difficult to 
see that Proposition~2.4 holds in the general case.
Proposition~2.4 is proved.  \qed

\section{On the proof of limit theorems for non-linear functionals 
of vector-valued stationary Gaussian random fields}

\medskip\noindent
First I recall the limit theorem problem we are interested in.

Let $X(p)=(X_1(p),\dots,X_d(p))$, $p\in{\mathbb Z}^\nu$, be 
a $d$-dimensional vector-valued Gaussian stationary field,
and let a function $H(x_1,\dots,x_d)$, 
$H\colon\;{\mathbb R}^d\to {\mathbb R}^1$,
of $d$ variables be given. Let us define with their help the 
random variables $Y(p)=H(X_1(p),\dots,X_d(p))$ for all 
$p\in{\mathbb Z}^\nu$, and introduce for all $N=1,2,\dots$
the normalized random sum
\begin{equation}
S_N=A_N^{-1}\sum_{p\in B_N} Y(p) \label{3.1}
\end{equation}
with an appropriate norming constant $A_N>0$, where
\begin{equation}
B_N=\{p=(p_1,\dots,p_\nu)\colon\; 0\le p_k<N \textrm { for all } 
1\le k\le\nu\}. \label{3.2}
\end{equation}

Let us also fix the vector-valued random spectral measure
$(Z_{G,1},\dots,Z_{G,d})$ on the torus $[-\pi,\pi)^\nu$ for which
$X_j(p)=\int e^{i(p,x)}Z_{G,j}(\,dx)$, $1\le j\le d$, 
$p\in{\mathbb Z}^\nu$. We are interested in the question 
what kind of limit theorems may hold for the normalized sums 
$S_N$ defined in~(\ref{3.1}) as $N\to\infty$ with an appropriate 
norming constants~$A_N$. Here we are interested in the case 
when the correlation functions $r_{j,j'}(p)=EX_j(0)X_{j'}(p)$,
$1\le j,j'\le d$, tend to zero slowly as $|p|\to\infty$. 
This means strong dependence of the random variables in 
the stationary random fields. In such cases, we can get 
limit theorems with a non-Gaussian limit.

We have studied the above problem in~\cite{4} for
scalar-valued stationary random fields, i.e., in the case
$d=1$, and we have proved some new kinds of limit theorems. 
Let me remark that at the same time M.~Taqqu also 
proved similar results with the help of a different 
method, see~\cite{11}. I do not discuss Taqqu's work, 
because here I am interested  in the question of how to 
generalize the method in~\cite{4} to prove limit theorems 
also for non-linear functionals of vector-valued 
stationary Gaussian random fields. 

In paper~\cite{8} and Section~2 of this work I discussed 
the notions and results we have to adopt some important
methods of~\cite{6} when we are working with multivariate
models. In this section, I explain how to generalize those
methods of~\cite{6} which lead to non-central limit
theorems when we are working with non-linear functionals
of vector-valued stationary Gaussian random fields. I
shall give the proof of the limit theorems in paper~\cite{9}
with the help of the results in this paper. 

In the first step of this discussion I rewrite the limit
problem we are interested in in a different form. Let us
observe that we have $X_j(p)=T_pX_j(0)$ with the shift
transform~$T_p$ for all $p\in{\mathbb Z}^\nu$ and
$1\le j\le d$, hence $Y(p)=T_pY(0)$, and we can rewrite the
sum in~(\ref{3.1}) in the form
\begin{equation}
S_N=A_N^{-1}\sum_{p\in B_N}T_pY(0). \label{3.3}
\end{equation}
As it will turn out the crucial point in the investigation
of our limit theorems is the study of limit theorems
in the special case when $Y(0)$ is a Wick polynomial, and
here we restrict our attention to this case.

Let us consider the case when $Y(0)$ is a Wick polynomial
of order~$k$ which has the form
\begin{equation}
Y(0)=\;\;\colon\!\!\!\!\!\!
 \sum_{\substack{(k_1,\dots,k_d)\\ k_1+\cdots+k_d=k}}
 a_{k_1,\dots,k_d}X_1(0)^{k_1}\cdots X_d(0)^{k_d}\colon \label{3.3a}
\end{equation}
with some real coefficients $a_{k_1,\dots,k_d}$.
Then by the corollary of Theorem~2.2 and the identities
$X_j((0)=\int {\mathbb I}_1(x)Z_{G,j}(\,dx)$, $1\le j\le d$, 
where ${\mathbb I}_1(\cdot)$
denotes the indicator function of the torus $[-\pi,\pi)^\nu$,
the random variable $Y(0)$ can be written in the form
$$
Y(0)= \!\!\!\! \sum_{\substack{(k_1,\dots,k_d)\\ k_j\ge0,\;1\le j\le d, \\
 k_1+\cdots +k_d=k}}\!\!\!\!
 a_{k_1,\dots,k_d} \int {\mathbb I}_1(x_1)\dots {\mathbb I}_1(x_k)\prod_{j=1}^d
\left(\prod_{t=k_1+\cdots+k_{j-1}+1}^{k_1+\cdots+k_j} \!\! Z_{G,j}(\,dx_t)\right),
$$
where for $j=1$ we define
$\prod\limits_{t=k_1+\cdots+k_{j-1}+1}^{k_1+\cdots+k_j} Z_{G,j}(\,dx_t)
=\prod\limits_{t=1}^{k_1} Z_{G,1}(\,dx_t)$, and if $k_j=0$ for some 
$1\le j\le d$, then the product 
$\prod\limits_{t=k_1+\cdots+k_{j-1}+1}^{k_1+\cdots+k_j}Z_{G_j}(\,dx_t)$
is omitted from this expression.

By Proposition~2.4 we can write
\begin{equation}
 T_pY(0)=\!\!\! \sum_{\substack{(k_1,\dots,k_d) \\
 k_j\ge0,\;1\le j\le d, \\  k_1+\cdots +k_d=k}} \!\!\!
 a_{k_1,\dots,k_d}\int e^{i(p,x_1+\cdots+x_k)}\prod_{j=1}^d
\left(\prod_{t=k_1+\cdots+k_{j-1}+1}^{k_1+\cdots+k_j} Z_{G.j}(\,dx_t)\right)
\label{3.4}
\end{equation}
for all indices $p\in{\mathbb Z}^\nu$.

We get by summing up formula~(\ref{3.4}) for all $p\in B_N$ 
with our choice of $Y(0)$ that
\begin{eqnarray*}
S_N=A_N^{-1}
\sum_{\substack{(k_1,\dots,k_d)\\ k_j\ge0,\;1\le j\le d, \\   k_1+\cdots +k_d=k}}
 a_{k_1,\dots,k_d}\int&&\prod _{l=1}^\nu 
\frac{e^{iN(x_1^{(l)}+\cdots+x_k^{(l)})}-1} 
{e^{i(x_1^{(l)}+\cdots+x_k^{(l)})}-1} \\ 
&&\qquad \prod_{j=1}^d
\left(\prod_{t=k_1+\cdots+k_{j-1}+1}^{k_1+\cdots+k_j} Z_{G,j}(\,dx_t)\right),
\end{eqnarray*}
where we write $x=(x^{(1)},\dots,x^{(\nu)})$ for all 
$x\in[-\pi,\pi)^\nu$. (The set $B_N$ was defined in~(\ref{3.2}).) 
I shall rewrite the above identity in a form more appropriate for
us, First I apply the change of variables
$y_l=Nx_l$, $1\le l\le n$. It yields that
\begin{equation}
S_N=
\sum_{\substack{(k_1,\dots,k_d)\\k_j\ge0,\;1\le j\le d, \\  k_1+\cdots +k_d=k}}
 \int h^N_{k_1,\dots,k_d}(y_1,\dots,y_k)\prod _{j=1}^d 
\left(\prod_{t=k_1+\cdots+k_{j-1}+1}^{k_1+\cdots+k_j} Z_{G^{(N)},j}(\,dy_t)\right),
\label{3.5}
\end{equation}
where
$$
h^N_{k_1,\dots,k_d}(y_1,\dots,y_k)=a_{k_1,\dots,k_d}\prod_{l=1}^\nu 
\frac{e^{i(y_1^{(l)}+\cdots+y_k^{(l)})}-1} 
{N(e^{i(y_1^{(l)}+\cdots+y_k^{(l)})/N}-1)} 
$$
is a function on $[-N\pi,N\pi)^\nu$, and 
$Z_{G^{(N)},j}(A)=N^{\nu/k}A_N^{-1/n}Z_{G,j}(\frac AN)$ is defined
for all measurable sets $A\subset [-N\pi,N\pi)^\nu$ and 
$j=1,\dots,d$. Here we use the notation 
$y_s=(y_s^{(1)},\dots,y_s^{(\nu)})$, $1\le s\le k$.  
Let us observe that $(Z_{G^{(N)},1},\dots,Z_{G^{(N)},d})$ is a 
vector-valued random spectral measure on the torus 
$[-N\pi,N\pi)^\nu$, corresponding to the matrix valued 
spectral measure $G^{(N)}=(G^{(N)}_{j,j'})$, $1\le j,j'\le d$, 
on the torus $[-N\pi,N\pi)^\nu)$, defined by the formula 
$$
G^{(N)}_{j,j'}(A)=N^{2\nu/k}A_N^{-2/n}G_{j,j'}\left(\frac AN\right), 
\quad 1\le j,j'\le d,
$$ 
for all measurable sets $A\subset[-N\pi,N\pi)^\nu$, where 
$G=(G_{j,j'})$, $1\le j,j'\le d$, is the matrix valued 
spectral measure of the original vector-valued stationary 
random field 
$X(p)=(X_1(p),\dots,X_d(p))$, $p\in{\mathbb Z}^\nu$.
On the other hand, 
$h^N_{k_1,\dots,k_d}\in{\mathcal K}_{k,j_1,\dots,j_k}
(G^{(N)}_{j_1,j_1},\dots,G^{(N)}_{j_k,j_k})$ with
$j_p=s$ if $k_1+\cdots+ k_{s-1}< p\le k_1+\cdots+k_s$, 
$1\le s\le d$. (For $s=1$ we define $k_1+\cdots+ k_{s-1}=0$.)

In formula~(\ref{3.5}) we have taken summation for the series
$(k_1,\dots,k_d)\in{\mathcal K}$, where
${\mathcal K}=\{(k_1,\dots,k_d)\colon k_s\ge0,\,1\le s\le d,\;
k_1+\cdots+k_d=k\}$, and in each term of the sum at its
right-hand side there was a product of the form
$\prod_{s=1}^k Z_{G^{(N)},j_s}(du_t)$ with a sequence
$(j_1,\dots,j_k)\in{\mathcal J}$, where
${\mathcal J}=\{(j_1,\dots,j_k)\colon
1\le j_1\le j_2\le\cdots \le j_k\le d\}$, and it depended
on the sequence $(k_1,\dots,k_d)$ which indices $j_1,\dots,j_k)$
appeared in the above product.

We want to rewrite expression~(\ref{3.5}) in a more pleasant
form. For this goal, we make the following observation.
There is a natural invertible map
$T\colon {\mathcal J}\to{\mathcal K}$, defined as
$T(j_1,\dots,j_k)=(k_1(j_1,\dots,j_k),\dots k_d(j_1,\dots,j_k))$
for all $(j_1,\dots,j_k)\in{\mathcal J}$, where for all indices
$1\le s\le d$ $k_s(j_1,\dots,j_k)$ equals the number of those
elements $j_p$  in the sequence $(j_1,\dots,j_k)$ for which $j_p=s$.
Its inverse is defined by the formula
$$
T^{-1}(k_1,\dots,k_d)=(j_1(k_1,\dots,k_d),\dots,j_k(k_1,\dots,k_d))
$$
for all $(k_1,\dots,k_d)\in{\mathcal K}$, where
$j_s(k_1,\dots,k_d)=\min\{u\colon k_1+\cdots+k_u\ge s\}$.

With the help of the above defined functions $k_s=k_s(j_1,\dots,j_k)$,
$1\le s\le d$, we can rewrite the identity in~(\ref{3.5}) in a
form more appropriate for us. In this new formula we take
summation for $(j_1,\dots,j_k)\in{\mathcal J}$ instead of
$(k_1,\dots,k_d)\in{\mathcal K}$. We get that
\begin{equation}
 S_N  =\sum_{\substack{(j_1,\dots,j_k),\\  1\le j_1\le\cdots\le j_k\le d}}
\int h^N_{j_1,\dots j_k}(y_1,\dots,y_k) 
Z_{G^{(N)},{j_1}}(\,dy_1)\dots Z_{G^{(N)},{j_k}}(\,dy_k) \label{3.5a}
\end{equation}
with
$$
h^N_{j_1,\dots,j_k}(y_1,\dots,y_k)
=a_{k_1(j_1,\dots,j_k),\dots,k_d(j_1,\dots,j_k)}\prod_{l=1}^\nu 
\frac{e^{i(y_1^{(l)}+\cdots+y_k^{(l)})}-1} 
{N(e^{i(y_1^{(l)}+\cdots+y_k^{(l)})/N}-1)}.
$$

Let us observe that
$$
\lim_{N\to\infty} h^N_{j_1,\dots,j_k}(y_1,\dots,y_k)
=h^0_{j_1,\dots,_k}(y_1,\dots,y_k)
$$
with the function
$$
h^0_{j_1,\dots,j_k}(y_1,\dots,y_k)
=a_{k_1(j_1,\dots,j_k),\dots,k_d(j_1,\dots,j_k)}\prod_{l=1}^\nu 
\frac{e^{i(y_1^{(l)}+\cdots+y_k^{(l)})}-1} 
{i(y_1^{(l)}+\cdots+y_k^{(l)})} 
$$
defined on ${\mathbb R}^{k\nu}$, and this convergence is uniform in all 
bounded subsets of ${\mathbb R}^{k\nu}$.

It is natural to expect that if the matrix valued spectral 
measures $G^{(N)}=(G^{(N)}_{j,j'})$, $1\le j,j'\le d$, converge to a 
matrix valued spectral measure $G^{(0)}=(G^{(0)}_{j,j'})$, $1\le j,j'\le d$,
defined on ${\mathbb R}^\nu$ in an appropriate way, then a limiting 
procedure in formula~(\ref{3.5a}) supplies the limit theorem 
$S_N\to S_0$ in distribution with
$$
S_0=\sum_{\substack{(j_1,\dots,j_k)\\ 1\le j_1\le \dots\le j_k\le d}}
\int h^0_{j_1,\dots j_k}(y_1,\dots,y_k) 
Z_{G^{(0)},{j_1}}(\,dy_1)\dots Z_{G^{(0)},{j_k}}(\,dy_k) 
$$
as $N\to\infty$, where $(Z_{G^{(0)},1},\dots,Z_{G^{(0)},d})$ is a
vector-valued random spectral measure on ${\mathbb R}^\nu$
corresponding to the matrix valued spectral measure $(G^{(0)}_{j,j'})$, 
$1\le j,j'\le d$. On the other hand, the convergence of the
spectral measures $G^{(N)}$ to the spectral measure $G^{(0)}$
is satisfied in many interesting models, for instance in the models
investigated in paper~\cite{9}.

\medskip
Next, I explain how to work out a precise method to prove limit
theorems on the basis of the above heuristic argument. In particular,
we are interested in the question of when the above sketched heuristic
argument can be carried out. In the scalar-valued case, this problem
was solved in Lemma~8.3 of \cite{6}. Here I prove the
vector-valued variant of this result.

In the formulation of Lemma~8.3 of~\cite{6}, we had to introduce 
a version of the notion of weak convergence of finite measures  
to a larger class of measures, to the class of so-called locally
finite measures. They are measures, whose restrictions to any
compact set are finite. Here I introduce a slight generalization of
the notion called vague convergence in~\cite{6} to the case
when we are working with complex measures of locally finite total
variation. In~\cite{6} we have worked with (positive)
measures. (The definition of complex measures on
${\mathbb R}^\nu$ with locally finite total variation was
explained in Section~4 of the paper~\cite{8}.)

\medskip\noindent
{\bf Definition of vague convergence of complex measures 
on ${\mathbb R}^\nu$ with locally finite total variation.} 
{\it Let $G_N$, $N=1,2,\dots$, be a sequence of complex  
measures on ${\mathbb R}^\nu$ with locally finite total variation. 
We say that the sequence $G_N$ vaguely converges to a complex
measure~$G_0$ on~${\mathbb R}^\nu$ with locally finite total 
variation (in notation $G_N\stackrel{v}{\rightarrow} G_0$) if
\begin{equation}
\lim_{N\to\infty}\int f(x)\,G_N(\,dx)=\int f(x)\,G_0(\,dx) \label{3.6}
\end{equation}
for all continuous functions~$f$ on ${\mathbb R}^\nu$ with a 
bounded support.}

\medskip
I shall take a sequence of sums of $k$-fold Wiener--It\^o 
integrals, and then I formulate Proposition~3.1 which states
that under some appropriate conditions these sums of random
integrals have a limit that can be expressed in an explicit
form. This result together with the representation of non-linear
functionals of vector-valued stationary Gaussian random fields
by means of multiple Wiener--It\^o integrals enable us to
prove limit theorems with a non-Gaussian limit for non-linear
functionals of vector-valued stationary Gaussian random fields.

\medskip
For all $N=1,2,\dots$ take a sequence of matrix valued 
non-atomic spectral measures $(G^{(N)}_{j,j'})$, $1\le j,j'\le d$, 
on the torus $[-A_N\pi,A_N\pi)^\nu$ with parameter $A_N$ such that
$A_N\to\infty$ as $N\to\infty$. Let us also take some functions 
$$
h^N_{j_1,\dots,j_k}(x_1,\dots,x_k)\in {\mathcal K}_{k,j_1,\dots,j_k}
={\mathcal K}_{k,j_1,\dots,j_k}(G^{(N)}_{j_1,j_1},\dots,G^{(N)}_{j_k,j_k})
$$
on the torus $[-A_N\pi,A_N\pi)^\nu$ for all $(j_1,\dots,j_k)$ with
$1\le j_s\le d$, $1\le s\le k$, and $N=1,2,\dots$. For all
$N=1,2,\dots$ fix a vector-valued random spectral measure 
$$
(Z_{G^{(N)},1},\dots,Z^{(N)}_{G^{(N)},d})
$$ 
on the torus $[-A_N\pi,A_N\pi)^\nu$ corresponding to the matrix 
valued spectral measure $(G^{(N)}_{j,j'})$, $1\le j,j'\le d$. 
Let us define with the help of these quantities the  
sums of $n$-fold Wiener--It\^o integrals
\begin{equation}
Z_N= \!\!\! \sum_{\substack{(j_1,\dots,j_k)\\ 
1\le j_s\le d \textrm{ for all }1\le s\le k}} \!\!\!
\int h^N_{j_1,\dots,j_k}(x_1,\dots,x_k)Z_{G^{(N)},{j_1}}(\,dx_1)
\dots Z_{G^{(N)},{j_k}}(\,dx_k),    \label{3.7}
\end{equation}
$N=1,2,\dots$. In the next result I show that under
appropriate conditions these random variables $Z_N$ converge
in distribution to a random variable $Z_0$ expressed
in the form of a sum of multiple Wiener--It\^o integrals.

\medskip\noindent
{\bf Proposition 3.1.} {\it For all $N=1,2,\dots$ let us
consider the sums of $k$-fold Wiener--It\^o integrals $Z_N$
defined in formula~(\ref{3.7}) with the help of certain
vector-valued random spectral measures $(Z_{G^{(N)},1},\dots,Z_{G^{(N)},d})$ 
corresponding to some non-atomic matrix valued spectral 
measures $(G^{(N)}_{j,j'})$, $1\le j,j'\le d$, defined on tori
$[-A_N,A_N)^\nu$ such that $A_N\to\infty$ as $N\to\infty$, and
functions 
$$
h^N_{j_1,\dots,j_k}(x_1,\dots,x_k)\in 
{\mathcal K}_{k,j_1,\dots,j_k}(G^{(N)}_{j_1,j_1},\dots,G^{(N)}_{j_k,j_k}).
$$
Let the coordinates $G^{(N)}_{j,j'}$, $1\le j,j'\le d$, of the 
matrix valued spectral measures $(G^{(N)}_{j,j'})$, $1\le j,j'\le d$, 
converge vaguely to the coordinates $G^{(0)}_{j,j'}$ of a non-atomic 
matrix valued spectral measure $(G^{(0)}_{j,j'})$, $1\le j,j'\le d$, on 
${\mathbb R}^\nu$ for all $1\le j,j'\le d$ as $N\to\infty$, and let 
$(Z_{G^{(0)},1},\dots,Z_{G^{(0)},d})$ be a vector-valued random 
spectral measure on ${\mathbb R}^\nu$ corresponding to the matrix
valued spectral measure $(G^{(0)}_{j,j'})$, $1\le j,j'\le d$. 
Let us also have some functions $h^0_{j_1,\dots,j_k}$
for all $(j_1,\dots,j_k)$ with $1\le j_s\le d$, $1\le s\le k$,
such that these functions and matrix valued spectral measures 
satisfy the following conditions.

\medskip
\begin{description}
\item[{\rm(a)}] The functions $h^0_{j_1,\dots,j_k}(x_1,\dots,x_k)$ 
are continuous on ${\mathbb R}^{k\nu}$ for all $1\le j_s\le d$, 
$1\le s\le k$,  and for all $T>0$ and indices 
$1\le j_s\le d$, $1\le s\le k$, the functions 
$h^N_{j_1,\dots,j_k}(x_1,\dots,x_k)$ converge uniformly to 
the function $h^0_{j_1,\dots,j_k}(x_1,\dots,x_k)$ on the 
cube $[-T,T]^{k\nu}$ as $N\to\infty$.
\item[{\rm(b)}] For all $\varepsilon>0$ there is some 
$T_0=T_0(\varepsilon)>0$ such that 
\begin{equation}
\int_{{\mathbb R}^{k\nu}\setminus [-T,T]^{k\nu}} |h^N_{j_1,\dots,j_k}(x_1,\dots,x_k)|^2
G^{(N)}_{j_1,j_1}(\,dx_1)\dots G^{(N)}_{j_k,j_k}(dx_k)
<\varepsilon^2 \label{3.8} 
\end{equation}
for all $1\le j_s\le d$, $1\le s\le k$, and $N=1,2\dots$ if
$T>T_0$.
\end{description}

\medskip
Then inequality~(\ref{3.8}) holds also for $N=0$, 
\begin{equation}
h^0_{j_1,\dots,j_k}\in{\mathcal K}_{k,j_1,\dots,j_k}
={\mathcal K}_{k,j_1,\dots,j_k}(G^{(0)}_{j_1,j_1},\dots G^{(0)}_{j_k,j_k}),
\label{3.9}
\end{equation}
the sum of random integrals  
\begin{equation}
Z_0=\!\! \sum_{\substack{(j_1,\dots,j_k)\\ 
1\le j_s\le d, \textrm{ for all }1\le s\le k}} \!\!
\int h^0_{j_1,\dots,j_k}(x_1,\dots,x_k)Z_{G^{(0)},{j_1}}(\,dx_1)
\dots Z_{G^{(0)},{j_k}}(\,dx_k) \label{3.10}
\end{equation}  
exists, and the random variables $Z_N$ defined in~(\ref{3.7})
satisfy the relation $Z_N\stackrel{\mathcal D}{\rightarrow} Z_0$ as 
$N\to\infty$, where $\stackrel{\mathcal D}{\rightarrow}$ 
denotes convergence in distribution.}

\medskip\noindent
{\it Remark 1.}\/ A complex measure $G^{(N)}_{j,j'}$ with
finite total variation defined on the torus $[-A_N\pi,A_N\pi)^\nu$
can be identified in a natural way with a complex
measure on ${\mathbb R}^\nu$ which is concentrated on its 
subset $[-A_N\pi,A_N\pi)^\nu$. We take this identification 
of $G^{(N)}_{j,j'}$ with a complex measure on ${\mathbb R}^\nu$ 
when we give meaning to formula~(\ref{3.6}) with 
$G_N=G^{(N)}_{j,j'}$ and $G_0=G^{(0)}_{j,j'}$ in the definition of
the vague convergence of the complex measures
$G^{(N)}_{j,j'}$ to $G^{(0)}_{j,j'}$ as $N\to\infty$.

\medskip\noindent
{\it Remark 2.}\/ In Proposition~3.1 we imposed two conditions
for the convergence of the sums of multiple Wiener--It\^o
integrals $Z_N$ defined in~(\ref{3.7}) to the sum of multiple
Wiener--It\^o integrals $Z_0$ defined in~(\ref{3.10}). First
we demanded that the spectral measures and kernel functions
appearing in the definition of the expressions $Z_N$ should
converge to the corresponding spectral measure and kernel
functions appearing in the definition of the expression
$Z_0$ in an appropriate way. We still imposed an additional
condition in part~(b) of Proposition~3.1. This is a
compactness type condition which implies that the essential
part of the random integrals in the definition of the random
variables $Z_N$ are essentially concentrated in a compact
set not depending on the parameter~$N$. I shall show in the
Appendix with the help of an example that condition~(b)
cannot be omitted from Proposition~3.1. If we drop it
Proposition~3.1 may not hold any longer. I shall also make
some additional remarks about Proposition~3.1.

\medskip
Proposition~3.1 is a multivariate version of Lemma~8.3
in~\cite{6}. I gave a simpler proof of the scalar-valued
result in Lemma~6.6 of~\cite{7}, and I shall adopt this proof. In
the proof of Proposition~3.1 we have to overcome some additional
difficulties, which arose because we are working with
vector-valued random fields. First I explain the method of the proof.

\medskip
In the first step of the proof, we show that relation~(\ref{3.9})
holds, and the random integrals appearing in the definition of~$Z_0$
really exist. The main step in the proof of this result is to show
that the measures $\mu^{(N)}_{j_1,\dots,j_k}$ defined at the start of
the proof weakly converge to a measure $\mu^{(0)}_{j_1,\dots,j_k}$.
First, we prove the vague convergence of these measures with the
help of the conditions at the beginning of~Proposition~3.1, and
then we show with the help of condition~(b) of this result that
weak convergence holds, too.

In the next step, we reduce the proof of Proposition~3.1 to
that of relation~(\ref{3.11}). This reduction contains the
statement of Proposition~3.1 in the special case when the
kernel functions
$$
h^N_{j_1,\dots,j_k}(x_1,\dots,x_k) \textrm{ and } h^0_{j_1,\dots,j_k}(x_1,\dots,x_k)
$$
in the random integrals in formulas~(\ref{3.7}) and~(\ref{3.10})
are replaced by
$$
h^0_{j_1,\dots,j_n}(x_1,\dots,x_k)\chi_T(x_1,\dots,x_k),
$$
where $\chi_T(x_1,\dots,x_k)$ is the indicator function of the
cube $[-T,T]^{k\nu}$.

The proof of this reduction is based on the application of
inequality~(5.6) in~\cite{8} which helps to show that 
the expressions $Z_N$ in~(\ref{3.7}) and~(\ref{3.10}) are
changing very little if we restrict the domain of integration
in the Wiener--It\^o integrals appearing in them to a cube
$[-T,T]^{k\nu}$ with a sufficiently large number~$T>0$ or if
we replace the kernel functions $h^N_{j_1,\dots,j_k}(x_1,\dots,x_k)$
in these random integrals by their limit
$h^0_{j_1,\dots,j_k}(x_1,\dots,x_k)$. We also exploit that the
weak convergence of distribution functions to a limit can
be reformulated by means of their characteristic functions
in an equivalent way.

Then we formulate a new, simpler version of the limit
relation~(\ref{3.11}) in formula~(\ref{3.16}) which implies
relation~(\ref{3.11}), hence Proposition~3.1, too. In this
version we replace the kernel functions
$h^0_{j_1,\dots,j_k}(x_1,\dots,x_k)\chi_T(x_1,\dots,x_k)$
in~(\ref{3.11}) by such simple functions which provide a
good approximation of them. (See Section~5 of~\cite{8} for
the definition of simple functions and of their properties
needed in our proof.) We want to find such simple functions
which provide a good approximation of the random integrals
appearing at the right-hand side of~(\ref{3.11}) if we replace
each kernel function in these integrals by the appropriate
element of them. First we want to guarantee only the weaker
statement that we get a good approximation by this
replacement in the special case when the expression at the
right-hand side of~(\ref{3.11}) is considered. On the other
hand, we prescribe an additional property of these simple
functions. We demand that they should be adapted to such a
regular system whose elements have boundaries with zero
$\mu_0$ measure with respect to an appropriate measure
$\mu_0$. We impose this extra condition because it will
imply that the replacement of the kernel functions in the
random integrals in~(\ref{3.11}) with the appropriate simple
functions provides a good approximation not only for the
expression on the right-hand side of this formula, where
we integrate with respect to the random spectral measures
$Z_{G^{(0)},j}$, but also for the expressions on the left-hand
side of this formula for large indices~$N$, where we integrate
with respect to the random spectral measures $Z_{G^{(N)},j}$.

The reduction of the proof of~(\ref{3.11}) to the proof
of~(\ref{3.16}) is done similarly to the reduction
of the proof of Proposition~3.1 to the proof of~(\ref{3.11})
with the help of formulas~(\ref{3.14}) and~(\ref{3.15}).
These relations are consequences of the properties of
the simple functions we chose to approximate the kernel
functions in the random integrals in~(\ref{3.11}). Then we
have to prove relation~(\ref{3.16}). At this point, some new
difficulties arise if we are working with vector (and not
scalar) valued random fields.

An essential point of the proof of~(\ref{3.16}) is to
show that under the conditions of Proposition~3.1
$G^{(N)}_{j,j'}(\Delta)\to G^{(0)}_{j,j'}(\Delta)$ as $N\to\infty$
for all $1\le j,j'\le d$ if the boundary of the set
$\Delta$ behaves nicely. The proof is fairly simple for
$j=j'$. In this case, we can apply some results about weak
convergence of measures. But the proof is more difficult for
pairs $(j,j')$ with $j\neq j'$. This difficulty arises,
because if $j\neq j'$, then $G^{(N)}_{j,j'}(\cdot)$ with a fixed
parameter~$N$ can be not a (real-valued, positive) measure.
To get a proof in this case we apply a special argument, where
we exploit that $G^{(N)}_{j,j'}$ is one of the coordinates of a
positive semidefinite matrix valued measure $G^{(N)}$.

If the limit behavior of the complex measures $G^{(N)}_{j,j'}$
with locally finite total variation is already known, then
relation~(\ref{3.16}) can be proved in a standard way. For all
$N=0,1,2,\dots$ we consider the random vector $Z_N({\mathcal D})$
we obtain if we restrict the random spectral measure $Z_{G^{N}}$
to the elements of the regular system~${\mathcal D}$ we are
working with. It can be proved that the distributions of
the random vectors $Z_N({\mathcal D})$ weakly converge to the
distribution of $Z_0({\mathcal D})$ as $N\to\infty$. As the
random variables $Y_N$ appearing in formula~(\ref{3.16}) are
polynomials of the arguments in the random vectors~$Z_N({\mathcal D})$,
the limit relation in~(\ref{3.16}) simply follows from the above
weak convergence.

\medskip\noindent
{\it Proof of Proposition 3.1.}\/ First I show that 
relation~(\ref{3.8}) holds also for $N=0$. To see this let
us first show that the measures $\mu^{(N)}_{j_1,\dots,j_k}$,
$N=1,2,\dots$, defined as
$$
\mu^{(N)}_{j_1,\dots,j_k}(A)=\int_A |h^N_{j_1,\dots,j_k}(x_1,\dots,x_k)|^2
G^{(N)}_{j_1,j_1}(\,dx_1)\dots G^{(N)}_{j_k,j_k}(dx_k), 
\; A\subset\mathbb R^{k\nu}, 
$$
converge vaguely to the locally finite measure
$\mu^{(0)}_{j_1,\dots,j_k}$ defined as
$$
\mu^{(0)}_{j_1,\dots,j_k}(A)
=\int_A |h^0_{j_1,\dots,j_k}(x_1,\dots,x_k)|^2
G^{(0)}_{j_1,j_1}(\,dx_1)\dots G^{(0)}_{j_k,j_k}(dx_k), 
\; A\subset\mathbb R^{k\nu}, 
$$
if $N\to\infty$.

Indeed, it follows from the vague convergence of the measures 
$G_{j,j}^{(N)}$ to $G^{(0)}_{j,j}$ as $N\to\infty$ and the continuity 
of the function $h^{(0)}_{j_1,\dots,j_n}$ that this relation holds if 
we replace the kernel function $|h^N_{j_1,\dots,j_k}(x_1,\dots,x_k)|^2$ 
by the kernel function $|h^0_{j_1,\dots,j_k}(x_1,\dots,x_k)|^2$ 
in the definition of the measures $\mu^{(N)}_{j_1,\dots,j_k}$. Then 
condition~(a) of Proposition~3.1 implies that this relation also 
holds with the original definition of the measures 
$\mu^{(N)}_{j_1,\dots,j_k}$. 

Next I state that the measure $\mu^{(0)}_{j_1,\dots,j_k}$ is finite, 
and the measures $\mu^{(N)}_{j_1,\dots,j_k}$ converge to it not only 
vaguely but also weakly. Indeed, condition~(b) implies that the
sequence of measures $\mu^{(N)}_{j_1,\dots,j_k}$ is compact with 
respect to the topology defining the weak convergence of 
finite measures, hence any subsequence of it has a convergent 
sub-subsequence. But the limit of such a sub-subsequence can be 
only its limit with respect to the vague convergence, i.e., it is 
$\mu^{(0)}_{j_1,\dots,j_k}$. This implies that $\mu^{(0)}_{j_1,\dots,j_k}$
is a finite measure, and the sequence of measures
$\mu^{(N)}_{j_1,\dots,j_k}$ converges also weakly to it.

Finally the properties of the functions $h^N_{j_1,\dots,j_k}$,
and their convergence to $h^0_{j_1,\dots,j_k}$ formulated in
condition~(a) imply that also the symmetry property
$h^0_{j_1,,\dots,j_k}(-x_1,\dots,-x_k)
=\overline{h^0_{j_1,,\dots,j_k}(x_1,\dots,x_k)}$ holds, hence
relation (\ref{3.9}) is valid, and the random integral $Z_0$
defined in (\ref{3.10}) is meaningful.

Next I reduce the proof of the relation 
$Z_N\stackrel{\mathcal D}{\rightarrow} Z_0$ to the proof of the 
following statement: 

Under the conditions of Proposition~3.1
\begin{eqnarray}
&&\sum_{\substack{(j_1,\dots,j_k)\\ 
1\le j_s\le d, \textrm{ for all }1\le s\le k}} \!\!
\int h^0_{j_1,\dots,j_k}(x_1,\dots,x_k) \chi_T(x_1,\dots,x_k)
\label{3.11}  \\
&& \qquad\qquad\qquad\qquad\qquad\qquad
Z_{G^{(N)},{j_1}}(\,dx_1)\dots Z_{G^{(N)},{j_k}}(\,dx_k)  \nonumber \\
&&\qquad\qquad \stackrel{\mathcal D}{\rightarrow}
\sum_{\substack{(j_1,\dots,j_k) \nonumber \\ 
1\le j_s\le d, \textrm{ for all }1\le s\le k}}
\int h^0_{j_1,\dots,j_k}(x_1,\dots,x_k) \chi_T(x_1,\dots,x_k) \\
&&\qquad\qquad\qquad\qquad\qquad\qquad\qquad 
Z_{G^{(0)},{j_1}}(\,dx_1)\dots Z_{G^{(0)},{j_k}}(\,dx_k), \nonumber 
\end{eqnarray}
as $N\to\infty$, where $\chi_T(x_1,\dots,x_k)$ is the indicator
function of the cube $[-T,T]^{k\nu}$. We make a small, not so
important technical restriction in the choice of the number
$T$~ in~(\ref{3.11}). Let me recall that for all all
vector-valued spectral measures~$G_N$ there is a finite measure~$\mu_N$
on ${\mathbb R}^\nu$ such that all coordinates $G_{j,j'}^{(N)}$ of
$G_N$, (more precisely their restrictions to any compact sets)
are such measures which are absolutely continuous measures with
respect to~$\mu_N$. (See Section~4 of~\cite{8}.) We fix such a
measure~$\mu_N$ for all $N=0,1,2,\dots$, and we shall call them
dominating measures. We shall work with such measures~$\mu_N$
in the proof of Proposition~3.1. We state formula~(\ref{3.11})
for all such $T>0$ for which the boundary of the cube
$[-T,T]^{k\nu}$ has zero measure with respect to the measure 
$\underbrace{\mu_0\times\cdots\times\mu_0}_{k \textrm{ times}}$.

To prove this reduction let us observe that by formulas~(5.6) 
in~\cite{8} and~(\ref{3.8}) 
\begin{eqnarray*}
&&\!\!\!\!\!\!\!\!\!
E\biggl[\int [1-\chi_T(x_1,\dots,x_k)]h^N_{j_1,\dots,j_k}(x_1,\dots,x_k)\\
&& \qquad\qquad\qquad\qquad 
Z_{G^{(N)},{j_1}}(\,dx_1)\dots Z_{G^{(N)},{j_k}}(\,dx_k)\biggr]^2 \\
&&\!\!\!\!\!\! \le k!\int_{{\mathbb R}^{k\nu}\setminus [-T,T]^{k\nu}}
|h^N_{j_1,\dots,j_k}(x_1,\dots,x_n)|^2
G^{(N)}_{j_1,j_1}(\,dx_1)\dots G^{(N)}_{j_k,j_k}(\,dx_k)<k!\varepsilon^2
\end{eqnarray*}
for all sequences $(j_1,\dots,j_k)$, $1\le j_s\le d$, $1\le s\le k$,
and $N=0,1,2,\dots$ if $T>T_0(\varepsilon)$.  Hence
\begin{eqnarray}
E\biggr[&&\sum_{\substack{(j_1,\dots,j_k) \nonumber \\ 
1\le j_s\le d \textrm{ for all }1\le s\le k}}
\int [1-\chi_T(x_1,\dots,x_k)]h^N_{j_1,\dots,j_k}(x_1,\dots,x_k)\\
&&\qquad\qquad
Z_{G^{(N)},{j_1}}(\,dx_1)\dots Z_{G^{(N)},{j_k}}(\,dx_k)\biggr]^2 
\le d^kk!\varepsilon^2 \label{3.12}
\end{eqnarray}
for all $N=0,1,\dots$ if $T>T_0(\varepsilon)$.

Since 
$G^{(N)}_{j,j}\stackrel{v}{\rightarrow} G^{(0)}_{j,j}$ for all $1\le j\le d$ 
as $N\to\infty$, hence for all $T>0$ there is some number $C(T)$
such that $G^{(N)}_{j,j}([-T,T])\le C(T)$ for all $N=1,2,\dots$ and 
$1\le j\le d$. Because of this estimate and the uniform convergence 
$h^N_{j_1,\dots,j_k}\to h^0_{j_1,\dots,j_k}$ on any cube $[-T,T]^{k\nu}$ 
we have
\begin{eqnarray*}
&& E\biggl[\int[h^N_{j_1,\dots,j_k}(x_1,\dots,x_k)-h^0_{j_1,\dots,j_k}
(x_1,\dots,x_k)]\chi_T(x_1,\dots,x_k) \\
&&\qquad\qquad\qquad\qquad\qquad\qquad\qquad 
Z_{G^{(N)},{j_1}}(\,dx_1)\dots Z_{G^{(N)},{j_k}}(\,dx_k)\biggr]^2   \\
&&\qquad\le k!\int_{[-T,T]^{k\nu}}
|h^N_{j_1,\dots,j_k}(x_1,\dots,x_k)-h^0_{j_1,\dots,j_k}(x_1,\dots,x_k)|^2\\
&&\qquad\qquad\qquad\qquad\qquad\qquad\qquad 
G^{(N)}_{j_1,j_1}(\,dx_1)\dots G^{(N)}_{j_k,j_k}(\,dx_k)<\varepsilon^2 
\end{eqnarray*}
for all $T>0$ and $(j_1,\dots,j_k)$, $1\le j_s\le d$, $1\le s\le k$, 
if  $N>N_1$ with some $N_1=N_1(T,\varepsilon)$. Hence
\begin{eqnarray}
E\biggr[&&\sum_{1\le j_1,\dots,j_k\le d}
\int[h^N_{j_1,\dots,j_k}(x_1,\dots,x_k)-h^0_{j_1,\dots,j_k}
(x_1,\dots,x_k)]
\label{3.13} \\
&&\qquad \chi_T(x_1,\dots,x_k)   
Z_{G^{(N)},{j_1}}(\,dx_1)\dots Z_{G^{(N)},{j_k}}(\,dx_k)\biggr]^2
\le d^k\varepsilon^2 \nonumber
\end{eqnarray}
for all $T>0$ if  $N>N_1$ with some $N_1=N_1(T,\varepsilon)$.

Let us define the quantities
\begin{eqnarray*}
U_N=U_N(T)=\sum_{1\le j_1,\dots,j_k\le d}
\int &&h^N_{j_1,\dots,j_k}(x_1,\dots,x_k)\chi_T(x_1,\dots,x_k) \\
&&\qquad Z_{G^{(N)},{j_1}}(\,dx_1)\dots Z_{G^{(N)},{j_k}}(\,dx_k),
\end{eqnarray*}
and
\begin{eqnarray*}
V_N=V_N(T)=\sum_{1\le j_1,\dots,j_k\le d}
\int &&h^0_{j_1,\dots,j_n}(x_1,\dots,x_k)\chi_T(x_1,\dots,x_k) \\
&&\qquad Z_{G^{(N)},{j_1}}(\,dx_1)\dots Z_{G^{(N)},{j_k}}(\,dx_k),
\end{eqnarray*}
$N=1,2,\dots$. We introduce the definition of $V_N=V_N(T)$ 
also for $N=0$, where we replace the spectral measures 
$Z_{G^{(N)},j}$, $N\ge 1$, $1\le j\le d$, by $Z_{G^{(0)},j}$, 
$1\le j\le d$, in the definition of~$V_N$. We can reduce the
proof of the relation $Z_N\stackrel{\mathcal D}{\rightarrow}Z_0$ 
to formula~(\ref{3.11}) in the following way. By
formula~(\ref{3.12}) we can state that 
\begin{eqnarray*}
|E(e^{itZ_N}-e^{itU_N})|&\le& E|(1-e^{it(Z_N-U_N)})|\le E|(t(Z_N-U_N)|\\
&\le& |t|(E(Z_N-U_N)^2)^{1/2}\le |t|(d^kk!)^{1/2}\varepsilon. 
\end{eqnarray*}
for all $t\in {\mathbb R}^1$ with the random variable $Z_N$ 
defined in~(\ref{3.7}) 
if $T>T_0$ and $N>N_0(\varepsilon)$. Similarly, 
$|E(e^{itU_N}-e^{itV_N})|
\le |t| (E(U_N-V_N)^2)^{1/2}\le |t|d^{k/2}\varepsilon$
for all $t\in {\mathbb R}^1$ and $N>N_0$ by inequality~(\ref{3.13}).
Besides, (\ref{3.12}) with $N=0$ implies that
$$
E|e^{itV_0}-Ee^{itZ_0}|\le |t|(E(Z_0-V_o)^2)^{1/2}\le |t|d^{n/2}\varepsilon
$$
for all $t\in{\mathbb R}^1$ if $T>T_0(\varepsilon)$, where $Z_0$ is
defined in~(\ref{3.10}) and $V_0$ after the definition of $V_N$ for
$N\ge1$. Finally, $Ee^{itV_N}\to Ee^{itV_0}$ for all $t\in {\mathbb R}^1$ 
if relation (\ref{3.11}) holds. These relations together imply that 
$|Ee^{itZ_N}-Ee^{itZ_0}|\le C(t)\varepsilon$ if $N>N_0(t,\varepsilon)$ 
with some numbers $C(t)$ and $N_0(t,\varepsilon)$. Since this
inequality holds for all $\varepsilon>0$, it implies that
$Z_N\stackrel{\mathcal D}{\rightarrow}Z_0$. (In formula~(\ref{3.11})
we imposed a condition on the parameter $T>0$. We demanded that 
the boundary of $[-T,T]^{k\nu}$ must have measure zero with 
respect to the product measure of $\mu_0$. It causes no problem 
that we can apply the above argument only for parameters~$T$ 
with this property.)

\medskip
We shall prove (\ref{3.11}) with the help of some statements 
formulated below. To formulate them let us first fix a 
number $T>0$ such that the boundary of the cube 
$[-T,T]^{k\nu}$ has zero measure with respect to the measure 
$\underbrace{\mu_0\times\cdots\times\mu_0}_{n\textrm{ times}}$.
Observe that 
$$
h^0_{j_1,\dots,j_k}(x_1,\dots,x_k)\chi_T(x_1,\dots,x_k)
\in{\mathcal K}_{k,j_1,\dots,j_k} (G^{(N)}_{j_1,j_1},\dots,G^{(N)}_{j_k,j_k})
$$
for all $T>0$ and $N=0,1,2,\dots$. I claim that for all 
$\varepsilon>0$ a regular system
${\mathcal D}={\mathcal D}(\varepsilon)
=\{\Delta_k,\;k=\pm1,\dots,\pm M\}$ can be constructed for
which all of its elements have zero measure with respect the
a dominating measure $\mu_0$, i.e. 
$\mu_0(\partial\Delta_k)=0$ for all $1\le |k|\le M$,
$\Delta_k\subset [-T,T]^\nu$ for all $1\le |k|\le M$,
and such that there exist some simple functions
$$
f^\varepsilon_{j_1,\dots,j_k}\in\hat{\mathcal K}_{n,j_1,\dots,j_k}
(G^{(0)}_{j_1,j_1},\dots,G^{(0)}_{j_n,j_k})
$$
indexed by the parameters $(j_1,\dots,j_k)$, $1\le j_s\le d$,
$1\le s\le k$, which are adapted to this regular system, and
satisfy the inequalities written down in the following two
formulas~(\ref{3.14} and~(\ref{3.15}).
\begin{eqnarray}
&&\int|h^0_{j_1,\dots,j_k}(x_1,\dots,x_k)\chi_T(x_1,\dots,x_k)
-f^\varepsilon_{j_1,\dots,j_k}(x_1,\dots,x_k)|^2  \nonumber \\
&&\qquad\qquad\qquad\qquad 
G^{(0)}_{j_1,j_1}(\,dx_1)\dots G^{(0)}_{j_k,j_k}(\,dx_k)
<\varepsilon^2 \label{3.14} 
\end{eqnarray}
for all $1\le j_s\le d$, $1\le s\le k$, and also 
\begin{eqnarray}
&&\int|h^0_{j_1,\dots,j_k}(x_1,\dots,x_k)\chi_T(x_1,\dots,x_k)
-f^\varepsilon_{j_1,\dots,j_k}(x_1,\dots,x_k)|^2  \nonumber\\
&&\qquad\qquad\qquad\qquad 
G^{(N)}_{j_1,j_1}(\,dx_1)\dots G^{(N)}_{j_k,j_k}(\,dx_k)
<\varepsilon^2 \label{3.15} 
\end{eqnarray}
for all $1\le j_s\le d$, $1\le s\le k$, and $N\ge N_0$
with some $N_0=N_0(\varepsilon,T)$.

I also claim that with such a choice of simple functions 
\begin{equation}
Y_N\stackrel{\mathcal D}{\rightarrow} Y_0
\label{3.16}
\end{equation}
as $N\to\infty$, where
\begin{eqnarray*}
Y_N&=&Y_N(\varepsilon,T) \\
&=& \!\!\!\!\!\!
\sum_{\substack{(j_1,\dots,j_k)\\ 
1\le j_s\le d \textrm{ for all }1\le s\le k}} \!\!\!\!\!
\int f^\varepsilon_{j_1,\dots,j_k}(x_1,\dots,x_k)Z_{G^{(N)},{j_1}}(\,dx_1)
\dots Z_{G^{(N)},{j_k}}(\,dx_k)
\end{eqnarray*}
for $N=0,1,2,\dots$.

Let us show that for all $\varepsilon>0$ there exists a
regular system~${\mathcal D}$ together with some simple
functions $f^\varepsilon_{j_1,\dots,j_k}$ adapted to it
which satisfy the desired properties.

Indeed, by Lemma~(5.2) of~\cite{8} for all $\varepsilon>0$
and parameters $(j_1,\dots,j_k)$, $1\le j_s\le d$, $1\le s\le k$,
there exists such a simple function $f^\varepsilon_{j_1,\dots,j_k}$
adapted to a regular system ${\mathcal D}_{j_1,\dots,j_k}$ in
such a way that this function $f^\varepsilon_{j_1,\dots,j_k}$
satisfies (\ref{3.14}), and the elements of
${\mathcal D}_{j_1,\dots,j_k}$ have boundaries with
zero $\mu_0$ measure. Let us make such a construction for all
parameters $(j_1,\dots,j_k)$. It can be seen that there is
a refinement ${\mathcal D}$ of the regular systems
${\mathcal D}_{j_1,\dots,j_k}$ such that all simple functions
$f^\varepsilon_{j_1,\dots,j_k}$ are adapted to it, and its elements
have boundaries with zero $\mu_0$ probability. (I omit the
details of this construction.) This regular system together
with the functions $f^\varepsilon_{j_1,\dots,j_k}$ adapted to it satisy
the desired requirements, because, as we shall see, their
properties imply that these functions satisfy not only~(\ref{3.14}),
but also (\ref{3.15}).

Relation~(\ref{3.11}) can be proved with the help of relations (\ref{3.14}), 
(\ref{3.15}) and (\ref{3.16}) similarly to the reduction of the relation
$Z_N\stackrel{\mathcal D}{\rightarrow} Z_0$ to formula~(\ref{3.11}).
Indeed, one gets from inequalities (\ref{3.14}), (5.6) in~\cite{8} 
and the definition of the quantities $V_N$ and $Y_0$, by applying an
argument similar to the proof of relation~(\ref{3.12}) that
$$
E(V_0-Y_0)^2\le k!k^d\varepsilon^2,
$$
and also
$$
E(V_N-Y_N)^2\le k^d k!\varepsilon^2
$$
if $N>N_0(\varepsilon,T)$ by~(\ref{3.15}) and~(5.6) in~\cite{8}. 

Then we can show with the help of these relations similarly 
to the reduction of the relation
$Z_N\stackrel{\mathcal D}{\rightarrow} Z_0$ to formula~(\ref{3.11})
that $|Ee^{itV_N}-Ee^{itY_N}|\le\varepsilon$, 
$|Ee^{itY_N}-Ee^{itY_0}|\le\varepsilon$, and
$|Ee^{it Y_0}-Ee^{it V_0}|\le\varepsilon$ if 
$N>N_0(\varepsilon,t,T)$ with some threshold index
$N_0(\varepsilon,t,T)$. Here in the first and third inequality
we apply the last two inequalities which were consequences
of (\ref{3.14}) and (\ref{3.15}), while the second inequality follows
from (\ref{3.16}). Since these relations hold
for all $\varepsilon>0$ they imply that 
$Ee^{itV_N}\to Ee^{itV_0}$ for all $t\in {\mathbb R}^1$ as 
$N\to\infty$, i.e. $V_N\stackrel{\mathcal D}{\rightarrow} V_0$ 
as $N\to\infty$, and this is formula~(\ref{3.11}) written with a 
different notation.

It remains to prove~(\ref{3.14}), (\ref{3.15}) 
and~(\ref{3.16}). We made such a construction of a
regular system and simple functions adapted to it
with the help of Lemma~5.2 in~\cite{8} which
satisfy~(\ref{3.14}). Then formula~(\ref{3.15}) follows
from some classical results about vague (and weak)
convergence of measures. Since we are working in the
proof of (\ref{3.15}) in a cube $[-T,T]^{k\nu}$ it is
enough to know the results about weak convergence to
carry out our arguments.

Let us first observe that since the restrictions of
the measures $G^{(N)}_{j,j}$ to $[-T,T]^\nu$ tend 
weakly to the restriction of the measure $G^{(0)}_{j,j}$
to the cube $[-T,T]^\nu$ as $N\to\infty$, we can
also say that the restrictions of the product
measures $G^{(N)}_{j_1,j_1}\times\cdots\times G^{(N)}_{j_k,j_k}$
to the cube $[-T,T]^{k\nu}$ converge weakly to 
the restriction of the product measure 
$G^{(0)}_{j_1,j_1}\times\cdots\times G^{(0)}_{j_k,j_k}$
on the cube $[-T,T]^{k\nu}$, as $N\to \infty$. On the other
hand, the function 
\begin{eqnarray*} 
&& H^0_{j_1,\dots j_k}(x_1,\dots,x_k) \\
&&\qquad= |h^0_{j_1,\dots,j_k}(x_1,\dots,x_k)\chi_T(x_1,\dots,x_k)-
f^\varepsilon_{j_1,\dots,j_k}(x_1,\dots,x_k)|^2
\end{eqnarray*}
is almost everywhere continuous with respect to the
measure $G^{(0)}_{j_1,j_1}\times\cdots\times G^{(0)}_{j_k,j_k}$.
By the general theory about convergence of measures these
properties imply that
\begin{eqnarray*}
&&\int H^0_{j_1,\dots j_k}(x_1,\dots,x_k)
G^{(N)}_{j_1,j_1}(\,dx_1)\dots G^{(N)}_{j_k,j_k}(\,dx_k) \\
&&\qquad \to\int H^0_{j_1,\dots j_k}(x_1,\dots,x_k)
G^{(0)}_{j_1,j_1}(\,dx_1)\dots G^{(0)}_{j_k,j_k}(\,dx_k)
\nonumber
\end{eqnarray*}
as $N\to\infty$. (Such a convergence is proved for 
probability measures for example in~\cite{2}.) A careful
analysis shows that this result remains valid for sequences 
of finite, but not necessarily probability measures. 
Let me remark that here we are working with (real, 
non-negative) measures. The last relation 
together with~(\ref{3.14}) imply~(\ref{3.15}).

To prove relation~(\ref{3.16}) first we show that
$G^{(N)}_{j,j'}(\Delta_k)\to G^{(0)}_{j,j'}(\Delta_k)$
as $N\to\infty$ for all $1\le j,j'\le d$ and
$\Delta_k\in{\mathcal D}$ with the regular system
${\mathcal D}$ we are working with. (Let me recall
that the boundary of all sets $\Delta_k\in{\mathcal D}$
has zero $\mu_0$ measure and hence also zero $G^{(0)}_{j,j'}$ measure.)

If $j=j'$ then this relation follows immediately from the
facts that $G^{(N)}_{j,j}\stackrel{v}{\rightarrow} G^{(0)}_{j,j}$,
$G^{(0)}_{j,j}(\partial \Delta_k)=0$ for all $1\le |k|\le M$,
and $G^{(N)}_{j,j}$ is a locally finite measure for all 
$N=0,1,2,\dots$. If $j\neq j'$, then we have to
apply a more refined argument, since in this case
we only know that $G^{(N)}_{j,j'}$ is a complex
measure with locally finite total variation.
In this case we will exploit that the matrix valued
measures $(G^{(N)}_{j.j'})$, $1\le j,j'\le d$, are positive
semidefinite. This implies that the Radon--Nikodym
derivatives $g^{(N)}_{j,j'}$ of the complex
measures $G^{(N)}_{j,j'}$ with respect to the dominating 
measure $\mu_N$ have the following property. For all
$N=0,1,2,\dots$ and $1\le j,j'\le d$ such that $j\neq j'$
the $2\times 2$ matrices
$$
g^{(N)}(x|j,j')=\left( \begin{array}{l}
g^{(N)}_{j,j}(x),\quad g^{(N)}_{j,j'}(x) \\
g^{(N)}_{j',j}(x),\quad ng^{(N)}_{j',j'}(x)
\end{array} \right)
$$
are positive semidefinite for $\mu_N$ almost all $x\in {\mathbb R}^\nu$.
Let us define for all non-negative functions $v(x)$, $x\in {\mathbb R}^\nu$ 
the vector $S(x|v)=(\sqrt{v(x)},\sqrt{v(x)})$. By exploiting that 
the matrices $g^{(N)}(x|j,j')$ are positive semidefinite we get that
\begin{eqnarray*}
&\int v(x)[G^{(N)}_{j,j}(\,dx)+G^{(N)}_{j,j'}(\,dx)
+G^{(N)}_{j',j}(\,dx)+G^{(N)}_{j',j'}(\,dx)]\\
&\qquad =\int S(x|v)g^{(N)}(x|j,j')S(x|v)^*\mu_N(\,dx)\ge0
\end{eqnarray*}
for all functions $v$ such that $v(x)\ge0$, $x\in {\mathbb R}^\nu$. 
Hence 
$H^{(N)}_{j.j'}=[G^{(N)}_{j,j}+G^{(N)}_{j,j'}+G^{(N)}_{j',j}+G^{(N)}_{j',j'}]$ 
is a locally finite measure on ${\mathbb R}^\nu$. Moreover 
$H^{(N)}_{j,j'}\stackrel{v}{\rightarrow} H^{(0)}_{j,j'}$ as $N\to\infty$.
This implies that 
$H^{(N)}_{j,j'}(\Delta_k)\to H^{(0)}_{j,j'}(\Delta_k)$, therefore
$G^{(N)}_{j,j'}(\Delta_k)+G^{(N)}_{j',j}(\Delta_k)\to 
G^{(0)}_{j,j'}(\Delta_k)+G^{(0)}_{j',j}(\Delta_k)$ 
as $N\to\infty$ for all $\Delta_k\in{\mathcal D}$.

We get similarly by working with the vectors
$R(x|v)=(\sqrt{v(x)},i\sqrt{v(x)})$ instead of the vectors
$S(x|v)=(\sqrt{v(x)},\sqrt{v(x)})$ for all functions 
$v(x)\ge0$, $x\in {\mathbb R}^\nu$, that
$K^{(N)}_{j.j'}=[G^{(N)}_{j,j}+iG^{(N)}_{j,j'}-iG^{(N)}_{j',j}+G^{(N)}_{j',j'}]$ 
is a a locally finite measure for all $N=0,1,2,\dots$, and
$K^{(N)}_{j,j}\stackrel{v}{\rightarrow} K^{(0)}_{j,j'}$ as $N\to\infty$.
Thus $K^{(N)}_{j,j'}(\Delta_k)\to K^{(0)}_{j,j'}(\Delta_k)$, therefore
$G^{(N)}_{j,j'}(\Delta_k)-G^{(N)}_{j',j}(\Delta_k)\to 
G^{(0)}_{j,j'}(\Delta_k)-G^{(0)}_{j',j}(\Delta_k)$ 
as $N\to\infty$ for all $\Delta_k\in{\mathcal D}$.
These relations imply that 
$G^{(N)}_{j,j'}(\Delta_k)\to G^{(0)}_{j,j'}(\Delta_k)$
for all $\Delta_k\in{\mathcal D}$.

\medskip
Let us define for all $N=0,1,2,\dots$ and our regular system
${\mathcal D}=\{\Delta_k,\;1\le |k|\le M\}$ the Gaussian random
vector
$$
Z_N({\mathcal D})=\left(\textrm{Re}\,Z_{G^{(N)},j}(\Delta_k),\,
\textrm{Im}\,Z_{G^{(N)},j}(\Delta_k),\quad |k|\le M, \;\; 1\le j\le d\right)
$$
I claim that the elements of the covariance matrices of
the random vectors 
$Z_N({\mathcal D})$ can be expressed by means of the numbers 
$G^{(N)}_{j,j'}(\Delta_k)$, $1\le |k|\le M$ and $1\le j,j'\le d$, 
and the covariance matrices of $Z_N({\mathcal D})$ converge 
to the covariance matrix of $Z_0({\mathcal D})$ as $N\to\infty$.
(In the proof of this statement I repeat some arguments applied
in the investigation of random spectral measures in Section~3
of~\cite{8}.)

To prove these relations observe that 
\begin{eqnarray*}
&&\textrm{Re}\,Z_{G^{(N)},j}(\Delta_k)=
\frac{Z_{G^{(N)},j}(\Delta_k)+\overline{Z_{G^{(N)},j}(\Delta_k)}}2, \\
&&\textrm{Im}\,Z_{G^{(N)},j}(\Delta_k)=
\frac{Z_{G^{(N)},j}(\Delta_k)-\overline{Z_{G^{(N)},j}(\Delta_k)}}{2i},
\end{eqnarray*}
and
$\overline{Z_{G^{(N)},j}(\Delta_k)}
=Z_{G^{(N)},j}(-\Delta_k)=Z_{G^{(N)},j}(\Delta_{-k})$.
(In the last identity we exploited also the properties of the regular
systems ${\mathcal D}$.) Also the properties of the regular systems
imply that if $\Delta_k,\Delta_l\in{\mathcal D}$, then we have either 
$\Delta_k\cap\Delta_l=\Delta_k$ or $\Delta_k\cap\Delta_l=\emptyset$.
The first identity holds if $l=k$ and the second one if $l\neq k$. 
Hence we have either
$EZ_{G^{(N)},j}(\Delta_k)\overline{Z_{G^{(N)},{j'}}(\Delta_l)}
=G^{(N)}_{j,j'}(\Delta_k)$
if $k=l$ or $EZ_{G^{(N)},j}(\Delta_k)\overline{Z_{G^{(N)},{j'}}(\Delta_l)}=0$
if $k\neq l$.  These relations imply that we can express all
covariances 
\begin{eqnarray*}
&&E\textrm{Re}\,Z_{G^{(N)},j}(\Delta_k)\textrm{Re}\,Z_{G^{(N)},{j'}}(\Delta_l), \quad
E\textrm{Re}\,Z_{G^{(N)},j}(\Delta_k)\textrm{Im}\,Z_{G^{(N)},{j'}}(\Delta_l) \\ 
&&\qquad \textrm{ and }
E\textrm{Im}\,Z_{G^{(N)},j}(\Delta_k)\textrm{Im}\,Z_{G^{(N)},{j'}}(\Delta_l)
\end{eqnarray*}
with the help of the quantities $G^{(N)}_{j,j'}(\Delta_k)$, 
$1\le j,j'\le d$, $1\le |k|\le M$. The convergence of the numbers
$G^{(N)}_{j,j'}(\Delta_k)$ to $G^{(0)}_{j,j'}(\Delta_k)$ also implies that
the covariance matrices of $Z_N({\mathcal D})$ converge to
the covariance matrix of $Z_0({\mathcal D})$ as $N\to\infty$.

The convergence of the covariance matrices of the Gaussian random
vectors $Z_N({\mathcal D})$ with expectation zero also implies that
the distributions of $Z_N({\mathcal D})$ converge weakly to the
distribution of $Z_0({\mathcal D})$ as $N\to\infty$. But then the
same can be told about any continuous functions of the coordinates
of the random vectors $Z_N({\mathcal D})$. Because of the definition
of the multiple Wiener--It\^o integrals of simple functions the random
variables $Y_N$ in formula~(\ref{3.16}) are polynomials, hence continuous
functions of the coordinates of the random vectors $Z_N({\mathcal D})$.
Besides, these polynomials do not depend on the parameter~$N$. Hence
the previous results imply that formula~(\ref{3.16}) holds. 
Proposition~3.1 is proved.  \qed

\medskip
To simplify the application of Proposition~3.1 we also prove 
the following lemma.

\medskip\noindent
{\bf Lemma 3.2.} {\it Let us have a sequence of matrix valued
spectral measures $(G^{(N)}_{j,j'})$, $N=1,2,\dots$, $1\le j,j'\le d$, 
on the torus $[-A_N\pi,A_N\pi]^\nu$ such that $A_N\to\infty$, 
and $G^{(N)}_{j,j'}\stackrel{v}{\rightarrow} G^{(0)}_{j,j'}$ with some 
complex measure $(G^{(0))}_{j,j'})$ with locally finite
total variation for all $1\le j,j'\le d$ as $N\to\infty$. 
Then $G^{(0)}=(G^{(0)}_{j,j'})$, $1\le j,j'\le d$, is a positive 
semidefinite matrix valued even measure on ${\mathbb R}^\nu$.}

\medskip\noindent
{\it Remark.}\/ Lemma 3.2 helps to show that in many interesting
cases the limit matrix $G^{(0)}=(G^{(0)}_{j,j'})$, $1\le j,j'\le d$,
of the spectral measures $G^{(N)}=(G^{(N)}_{j,j'})$, $1\le j,j'\le d$, 
$N=1,2,\dots$, in Proposition~3.1 is the spectral measure of a
generalized stationary Gaussian random field. Indeed, in
Theorem~4.1 of~\cite{8} it was shown that a positive 
semidefinite matrix valued measure on $\mathbb R^\nu$
whose distribution is moderately increasing at infinity is 
the spectral measure of such a random field. (See Section~4 
of~\cite{8} for the definition of these notions.) So by 
Lemma~3.2 to prove that $G^{(0)}$ is the spectral measure of 
a generalized stationary Gaussian random field it is enough 
to show that the distribution of $G^{(0)}$ is moderately 
increasing. 

\medskip\noindent
{\it Proof of Lemma 3.2.}\/ We have to show that
$(G^{(0)}_{j,j'})$, $1\le j,j'\le d$, is a positive 
semidefinite matrix valued measure. To do this take 
a vector $v(x)=(v_1(x),\dots,v_d(x))$ whose coordinates 
$v_k(x)$, $1\le k\le d$, are continuous functions with 
compact support. We have 
\begin{equation}
\lim_{N\to\infty} \sum_{j=1}^d \sum_{j'=1}^d 
\int v_j(x)v_{j'}(x)G^{(N)}_{j,j'}(\,dx)
=\sum_{j=1}^d\sum_{j'=1}^d 
\int v_j(x)v_{j'}(x)G^{(0)}_{j,j'}(\,dx)\ge0. \label{3.17}
\end{equation}
The identity in~(\ref{3.17}) holds, since
$G^{(N)}_{j,j'}\stackrel{v}{\rightarrow} G^{(0)}_{j,j'}$ for all 
$1\le j,j'\le d$. The inequality at the end of (\ref{3.17}) 
also holds, because $(G^{(N)}_{j,j'})$, $1\le j,j'\le d$, is 
a positive semidefinite matrix valued measure for all 
$N=1,2,\dots$, and this implies that the left-hand side 
of~(\ref{3.17}) is non-negative for all $N=1,2,\dots$. Thus 
we got that if $g^{(0)}_{j,j'}(x)$ is the Radon--Nikodym 
derivative of $G^{(0)}_{j,j'}$ with respect to some 
dominating measure $\mu_0$ in the point $x\in {\mathbb R}^\nu$ 
for all $1\le j,j'\le d$, we take the $d\times d$ 
matrix $g^{(0)}(x)=(g^{(0)}_{j,j'}(x))$, $1\le j,j'\le d$, and 
the coordinates of the vector 
$v(x)=(v_1(x),\dots,v_d(x))$ are continuous functions 
with compact support, then
$$
\int v(x)g^{(0)}(x)v^*(x)\mu_0(\,dx)\ge0.
$$
In the proof of Theorem~2.2 of~\cite{8} we have shown that 
this relation implies that $(G^{(0)}_{j,j'})$, $1\le j,j'\le d$, 
is a positive semidefinite matrix valued measure. 

We still have to show that the complex  measure 
$G^{(0)}_{j,j'}$ with locally finite  variation is even for all
$1\le j,j'\le d$. To do this fix a pair $j,j'$ of indices, 
$1\le j,j'\le d$, and define for all $N=0,1,2,\dots$ the 
complex measure $(G')^{(N)}_{j,j}$ by the relation
$(G')^{(N)}_{j,j'}(A)=\overline{G^{(N)}_{j,j'}(-A)}$ for all bounded, 
measurable sets $A\subset {\mathbb R}^\nu$. It is not difficult to
see that not only 
$G^{(N)}_{j,j'}\stackrel{v}{\rightarrow} G^{(0)}_{j,j'}$,
but also $(G')^{(N)}_{j,j'}\stackrel{v}{\rightarrow} (G')^0_{j,j'}$ 
as $N\to\infty$. The evenness of the measures $G^{(N)}_{j,j'}$ 
for $N=1,2,\dots$ means that $G^{(N)}_{j,j'}=(G')^{(N)}_{j,j'}$ for
all $N=1,2,\dots$. By taking the limit $N\to\infty$ we
get that $G^{(0)}_{j,j'}=(G')^{(0)}_{j,j'}$. This means that 
$G^{(0)}_{j,j}$ is an even complex measure with 
locally finite variation. Lemma~3.2 is proved. \qed

\appendix
\section{On the results in Section 2.}
\label{a}

I wrote an Appendix consisting of two parts. In the first part I
discuss some questions related to Section~2 while in the second part
some questions related to Section~3.

In the first part, I write about the idea of the proof of It\^o's
formula together with the introduction of Wick polynomials. I
also explain how It\^o's formula is applied in the proof of the
other results in Section~2.

In the second part, I write about the investigation of limit
problems for non-linear functionals a stationary Gaussian
random fields and the content of Proposition~3.1. Here I want
to explain the role of the compactness type condition~(b) in
this proposition. I shall do this with the help of an example.

\medskip
First I try to explain the idea behind the proof of It\^o's
formula. To do this first I consider its one-dimensional
version. In that case we want to prove the following statement.
Let $\varphi(x)$, $\varphi(-x)=\overline{\varphi(x)}$, be such
a function for which $\int|\varphi(x)|^2 G(\,dx)=1$ with a
spectral measure $G$, and consider a random spectral measure
$Z_G$ corresponding to the spectral measure $G$. Then
$Y=\int\varphi(x)Z_G(\,dx)$ is a standard normal random variable,
and the identity
\begin{equation}
H_n\left(\int\varphi(x)Z_G(\,dx)\right)=\int \varphi(x_1)\cdots
\varphi(x_n) Z_G(\,dx_1) \dots Z_G(\,dx_n).  \label{A1}
\end{equation}
holds for all $n\ge1$.
Actually,  It\^o's formula is a more general result, but here
it will be enough to consider this special case.

It is proved in the general theory that the above-defined
random variable~$Y$ has standard normal distribution.
Formula~(\ref{A1}) is proved by induction with respect to~$n$.
In this induction, we apply the recursion formula~(\ref{2.2})
for Hermite polynomials and the diagram formula for the
product of multiple Wiener--It\^o integrals. We exploit
that they ``fit to each other''.

Formula~(\ref{A1}) clearly holds for $n=1$. To prove it
for $n$ if we know it for $m<n$ we rewrite the left-hand
side of~(\ref{A1}) with the help of the recursion
formula~(\ref{2.2}). We rewrite the random integral
$$
\int \varphi(x_1)\cdots\varphi(x_n) Z_G(\,dx_1) \dots Z_G(\,dx_n)
$$
at the right hand of~(\ref{A1}) by means of the identity
that we get by applying the diagram formula for the product
$$
\int \varphi(x_1)\cdots\varphi(x_{n-1}) Z_G(\,dx_1) \dots Z_G(\,dx_{n-1})
\int\varphi(x)Z_G(\,dx).
$$
(I remark that Proposition~5.1 of~\cite{6} yields a generalization
of the formula we get in such a  way.) Then some calculations with
the help of these formulas and the inductive hypothesis yield
the proof of formula~(\ref{A1}) for~$n$.

It\^o's formula for vector-valued stationary Gaussian random
fields can be proved by an appropriate adaptation of the above
argument. In the proof, we apply a useful special case of the
diagram formula for vector-valued stationary Gaussian random
fields presented in~\cite{8}. It is formulated in the
corollary of Theorem~6.1 in\cite{8}.

On the other hand, we need a new identity instead of formula~(2.2)
in the proof, and we can formulate it with the help of Wick
polynomials. This was the reason for their introduction in this
paper. They are defined in Section~2 with the help of some results
in~\cite{8}. First we have to understand that this definition is
correct. Namely, we have to show that the Wick polynomial
$\colon\!P(\xi_1,\dots,\xi_m)\colon$ of a homogeneous polynomial
$P(\xi_1,\dots,\xi_m)$ of order $n$ depends only on the random
variables $\xi_1,\dots,\xi_m$, although in its definition we
applied a projection to a Hilbert space ${\mathcal H}_n$ which
may depend on other random variables, too.

The result of Theorem~2B implies the correctness of this definition.
It states that the value of the Wick polynomial does not change
if we take projection to the Hilbert space
${\mathcal H}_n(\xi_1,\dots,\xi_m)$ introduced before the formulation
of this result instead of the projection to ${\mathcal H}_n$.
The definition of this new Hilbert space is similar to that of
${\mathcal H}_n$, the only difference is that here we work only
with the random variables $\xi_1,\dots,\xi_m$.

The proof of Theorem~2B exploits the following property of
Gaussian random vectors. If some coordinates of a Gaussian
random vector are uncorrelated, then they are also independent.
This implies that the elements of the underlying Gaussian
random field can be decomposed as $X_t=\eta_{1,t}+\eta_{2,t}$,
$t\in T$, in such a way that $\eta_{1,t}$ is a linear combination
of the random variables $\xi_1,\dots,\xi_m$, while $\eta_{2,t}$ is
uncorrelated, hence independent of them. The proof of Theorem~2B
is based on this fact. I omit the details of the proof.

Theorem 2B implies in particular that $\colon\!\xi^n\colon=H_n(\xi)$
if $\xi$ is a standard normal random variable. Corollary~2C describes
a deeper relation between Hermite and Wick polynomials. This can be
exploited. For instance, the identity~(\ref{2.3}) formulated in
Proposition~2.1 can be proved with its help and formula~(\ref{2.2})
about Hermite polynomials. This identity plays an important role
in the proof of It\^o's formula.

The proof of It\^o's formula for vector-valued stationary Gaussian
random fields is made with the help of the identity~(\ref{2.3})
for Wick polynomials and the Corollary of Theorem~6.1 in~\cite{8}.
(Theorem~6.1 in~\cite{8} is the diagram formula for vector-valued
stationary Gaussian random fields.) It is a natural adaptation of
the previously discussed proof in the scalar-valued case.

It is easy to deduce from Theorem~2.2 its Corollary. In this
Corollary the Wick polynomials of such homogeneous polynomials
are considered whose arguments are elements of one of the Hilbert
spaces ${\mathcal H}_{1,j}$, $1\le j\le d$. In the Corollary of
Theorem~2.2 such expressions are expressed in the form of
a sum of multiple Wiener--It\^o integrals. The class of
homogeneous polynomials considered in the Corollary of
Theorem~2.2 is fairly large. This fact is exploited in the
proof of Proposition~2.3, which states that the set of all
finite sums of $n$-fold Wiener--It\^o integrals constitute
an everywhere dense class of functions in~${\mathcal H}_n$.

In the last result of Section~2, in Theorem~2.4 a random variable,
defined in formula~(\ref{2.6}) in the form of a multiple
Wiener--It\^o integral is considered, and its shift transforms
are given in formula~(\ref{2.8}). This is an important result,
and its proof is based also on the It\^o formula. The validity of
formula~(\ref{2.8}) can be checked first for the shifts of
one-fold Wiener--It\^o integrals.
Then it can be proved with the help of the It\^o formula for such
multiple Wiener--It\^o integrals whose kernel functions have the
special form
$$
h(x_1,\dots,x_n)=\varphi_1(x_1)\cdots\varphi_n(x_n).
$$
After this Proposition~2.4 can be proved in the general case
by means of a standard method.



\section{On the results in Section 3.}
\label{B}

At the beginning of Section~3 I formulated a limit problem. I
considered a vector-valued stationary Gaussian random field $X(p)$,
$p\in{\mathbb Z}^\nu$, defined a Wick polynomial of order~$k$ of
the coordinates of the vector $X(0)$ in~(\ref{3.3a}), and I was
interested in a limit problem for the expressions~$S_N$ defined
in~(\ref{3.3}). These expressions are normalized partial sums
whose elements are shifts~$T_pY(0)$ of the random variable $Y(0)$.
This problem is a multivariate version of the problem studied
in~\cite{4}.

The expressions~$S_N$ can be rewritten in an interesting simple
form with the help of It\^o's formula that enables us to rewrite
$Y(0)$ in the form of a sum of multiple Wiener--It\^o integrals,
and of Proposition~2.4 that gives a useful formula for its
shifts $T_pY(0)$. I could express with their help and with an
appropriate rescaling of the random integrals we
are working with the expression~$S_N$ in the form given
in~(\ref{3.5a}). I applied this rescaling to get such an
expression~in(\ref{3.5a}), where the kernel functions of the
multiple Wiener--It\^o integrals have a limit as $N\to\infty$.
This formula suggests that if the matrix valued spectral
measures $G^{(N)}$ have a limit, then we can get the limit
of the random variables $S_N$ by means of a natural
limiting procedure. Maybe, this can be done only if some
additional not too restrictive conditions are satisfied.
Proposition~3.1 gives a useful sufficient condition for
the right to carry out such a limiting procedure, and
to get a limit theorem in this way.

In the formulation of Proposition~3.1 some random variables
$Z_N$, $N=0,1,2,\dots$, defined in~(\ref{3.7}) and~(\ref{3.10})
are considered. First it is shown that under the conditions of
Proposition~3.1 these random variables exist (the multiple
integrals appearing in their definition are well-defined), and
then also the convergence
$Z_N\stackrel{\mathcal D}{\rightarrow}Z_0$ is proved.

The first conditions of Proposition~3.1 are natural. They demand
that the kernel functions $h^N_{j_1,\dots,j_k}(x_1,\dots,x_k)$
of the random integrals in $Z_N$ and the matrix valued spectral
measures $G^{(N)}$ converge to $h^0_{j_1,\dots,j_k}(x_1,\dots,x_k)$
and $G^{(0)}$ in an appropriate way. (Actually, there is also a
hidden condition here. It is the condition that the functions
$h^0_{j_1,\dots,j_k}(x_1,\dots,x_k)$ must be continuous. In the
scalar-valued version of this result, in Proposition~8.3
of~\cite{6} a slightly weaker continuity condition is imposed.
Here we do not discuss the question how the continuity property
in Proposition~3.1 can be weakened.)

On the other hand, part~(b) of Proposition~8.3 contains
an additional condition that deserves special attention. We
can understand its role better by considering the application
of Proposition~3.1 in the proof of the main result in~\cite{9}
or the application of its scalar-valued version in the proof
of Theorem~8.2 of~\cite{6}.

In Theorem~8.2 of~\cite{6} a scalar-valued stationary Gaussian
random field $X(p)$, $p\in{\mathbb Z}^\nu$, is considered. The
random variables $Y(p)=H_k(X(p))=T_pY(0)$ are introduced, where
$H_k(\cdot)$ is the Hermite polynomial of order~$k$, and the
normalized partial sums $S_N$ are defined by formula~(\ref{3.3})
with these random variables $Y(p)$. A non-central limit theorem
is proved with normalizing constants $A_N=N^{\nu-k\alpha/2}L(N)^{k/2}$
if the corelation function $r(p)=EX(0)X(p)$ satisfies the relation
$r(p)= |p|^{-\alpha} a\left(\frac{p}{|p|}\right)L(|p|)$ with some
$0<\alpha<\frac\nu k$, where $L(\cdot)$ is a slowly varying
function at infinity, and $a(\cdot)$ is a function on the unit
sphere of ${\mathbb R}^\nu$. It shows the dependence of the
correlation function $r(p)$ on the direction of the vector~$p$.

Paper~\cite{9} contains a multivariate version of this result.
Here a vector-valued stationary Gaussian random field
$X(p)=(X_1(p),\dots,X_d(p))$, $p\in{\mathbb Z}^\nu$, is considered,
and a limit theorem is proved for the random variables $S_N$,
defined in~(\ref{3.3}) and~(\ref{3.3a}) under appropriate condition.
A condition, similar to the condition of the correnlation function
in Theorem~8.2 of~\cite{6} is imposed on the correlation function
$r_{j,j'}(p)=EX_j(0)X_{j'}(p)$ which is described in formula~(1.3)
of~\cite{8}. A non-central limit theorem with normalizing
constants $A_N=N^{\nu-k\alpha/2}L(N)^{k/2}$ is proved if this
condition holds for the correlation function with exponent
$0<\alpha<\frac \nu k$.

It is worth understanding why the condition $\alpha<\frac\nu k$
in the exponent of the formula expressing the decrease of the
correlation function is needed in the proof of these results. In
the first step of these proofs, we have to describe the asymptotic
behavior of the spectral measure of the underlying stationary
Gaussian random field. In the scalar-valued this is done in
Lemma~8.2 of~\cite{6}. It describes the limit behavior of
the appropriately rescaled versions~$G_N$ of the spectral
measure $G$ of the stationary Gaussian random field we are
working with. These measures are defined by the identity
$G_N(A)=\frac{N^\alpha}{L(N)}G\left(\frac AN\right)$ for all
measurable sets~$A$. In this lemma, it is proved that the
measures~$G_N$ have a vague limit $G_0$ as $N\to\infty$,
and the limit measure has the homogeneity property
$G_0(A)=t^{-\alpha}G_0(tA)$ for all measurable sets~$A$ and
$t>0$. There is a similar result also in the case of
vector-valued stationary Gaussian fields.

The above results imply that the spectral measures and 
kernel functions in the representation of $S_N$ in
formula~(\ref{3.5a}) satisfy the starting conditions of
Proposition~3.1 if the correlation function of the underlying
stationary Gaussian field has such an asymptotic behavior at
the infinity as it is demanded in the above-mentioned results.
Moreover, they are satisfied if the correlation function
satisfies the desired relation with arbitrary ``homogeneity
parameter'' $\nu>\alpha>0$.

One would like to understand where we exploited the condition
$\alpha<\frac\nu k$, and what can be told if this condition
is violated. (The number $k$ was the order of the Wick polynomial
we were working with.) This question is related to condition~(b)
in Proposition~3.1. It is proved that this condition is satisfied
under the additional condition $\alpha<\frac\nu k$. (This is proved
in~\cite{4} or~\cite{6} in the scalar and in~\cite{9} in the
vector-valued case.) On the other hand, it can be proved that if
the asymptotic formula prescribed for the correlation function
satisfies the prescribed asymptotic relation with
$\alpha\ge\frac\nu k$, then relation~(\ref{3.9}) does not
hold, and as a consequence, the definition of the random
variable~$Z_0$ in~(\ref{3.10}) is incorrect.

We have more detailed knowledge of the behavior of the random
sum~$S_N$ defined in~(\ref{3.3}) if $\alpha>\frac\nu k$. It is known
that it satisfies the central limit theorem with the standard
normalization $N^{\nu/2}$. This follows from the central limit
theorem proved in~\cite{3} in the scalar and in Theorem~4
of~\cite{1} in the vector-valued case. One only has to check
that the conditions of these results are satisfied in this
case, and this can be done by calculating the necessary covariances.
A similar central limit theorem also holds if $\alpha=\frac\nu k$,
but in this case, it may happen that the norming constant is
$N^{\nu/2}L(N)$ with a slowly varying function $L(N)$ tending to
infinity as $N\to\infty$.

The above-discussed results suggest the following heuristic
picture about a generalized version of the results discussed
in the Appendix.

Let us have a stationary Gaussian random field $X(p)$,
$p\in{\mathbb Z}^\nu$ and a non-linear functional $Y$ of this
random field. Take the shifts $Y(p)=T_pY$, $p\in{\mathbb Z}^\nu$,
of this non-linear functional, and consider their normalized
sums~$S_N$ defined in formula~(\ref{3.1}). We are interested
in what kind of limit theorem holds for this sequence $S_N$
as $N\to\infty$ if we choose the norming constants $A_N$
appropriately. In Section~3 we considered a special case of
this problem and proved by applying a method that it can be
reformulated to the problem about the limit theorem for a
sequence~$Z_N$ defined in~(\ref{3.7}). Such a sequence
has a limit if the kernel functions and spectral measures
in the definition of~$Z_N$ behave nicely. A similar
reformulation of the above-mentioned limit problem
is possible in a more general case, only different kernel
functions appear in the definition of the random integrals
in~(\ref{3.7}). Proposition~3.1 may help in the study of the
limit behavior of the random variables~$Z_N$ defined
in~(\ref{3.7}). It turned out that condition~(b) of
Proposition~3.1 is an important condition of this result.
Let us understand its role better.

Condition~(b) of Proposition~3.1 is a compactness type condition
formulated in~(\ref{3.8}). Let us consider the integrals in it
when we integrate on the whole space~${\mathbb R}^\nu$. If the
values of these integrals tend to infinity as $N\to\infty$,
then for large~$N$ the essential part of the random integrals
in~(\ref{3.7}) comes from a region that contains vectors with
a very big norm in~${\mathbb R}^{k\nu}$. It is natural to expect
that in such cases the random variables~$Z_N$ satisfy the central
limit theorem with the classical norming constants $A_N=N^{\nu/2}$
under very general conditions. Such a result is proved beside
the above-mentioned papers~\cite{1} and~\cite{3} also in the
book~\cite{10}. On the other hand, Proposition~3.1 implies the
existence of a non-Gaussian limit, expressed by means of a sum
of multiple Wiener--It\^o if the the spectral measures and the
kernel functions have a limit, and condition~(b)
of Proposition~3.1 holds.

\bigskip\noindent
{\it Acknowledgements.} This research was supported by
the Hungarian Foundation NKFI--EPR No. K-125569. 

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\end{document}






 
Finally I would mention a result of paper~\cite{5a} which deserves
special attention. Here a two-dimensional vector-valued stationary
Gaussian random sequenc $(X_m,Y_m)$, $m=0,\pm1,\pm2,\dots$,
$EX_m=EY_m=$, $EX_m^2=EY_m^2=1$ is considered with correlation
function satisfies the relations $r_1(m)=EX_0X_m\sim |m|^{-\beta_1}$,
$r_2(m)=EY_0Y_m\sim  |m|^{-\beta_2}$ as $m\to\infty$ together with
some Hermite polynomials $H_{\nu_1}(x)$ and $H_{\nu_2}(x)$ such that
$\nu_1\beta_1<1$ and $\nu_2\beta_2>1$. It is known that the
normalized sums $Z_1^n=n^{-(1-\nu_1\beta_1/2}\sum_{m=0}^{n-1}H_{\nu_1}(X_m)$
and $Z_2^n=n^{-1/2}\sum_{m=0}^{n-1}H_{\nu_2}(Y_m)$ converge in distribution
to some random variables $Z^*_1$ and $Z^*_2$, where  $Z^*_2$ has
normal distribution, and $Z^*_1$ can be defined by means of a
$\nu_1$-fold Wiener--It\^o integral. (See~\cite{3} and~\cite{4}.) 



\bibitem{5a} Ho, H. C. and Sun T. C. (1990) Limit distributions of
non-linear vector functions of stationary Gaussian processes.
Ann. Probab. {\bf 18} 1159--1173
