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\begin{document}

\title{Wiener--It\^o integral representation in vector valued
Gaussian stationary random fields}

\author{P\'eter Major \\
Mathematical Institute of the Hungarian 
Academy of Sciences \\
Budapest, P.O.B. 127 H--1364, Hungary, e-mail:major@renyi.hu}

%\plainfootnote{} {The author got support from the Hungarian 
%Foundation NKFI--EPR No. K-125569.}

\maketitle
\noindent
{\it Abstract:}\/ The subject
of this work is the multivariate generalization of the 
theory of multiple Wiener--It\^o integrals. In the scalar 
valued case this theory was described in paper~\cite{11}. Our 
proofs apply the technique of this work, but in the proof of 
some results new ideas were needed. The motivation for this 
study was a result in paper~\cite{1} of Arcones where he 
formulated the multivariate version of a non-central limit 
theorem for non-linear functionals of Gaussian stationary 
random fields presented in paper~\cite{6}. We found the proof in 
paper~\cite{1} incomplete and wanted to give a full proof. 
We did it in paper~\cite{13}, but in that proof we needed a 
detailed description of the properties of non-linear 
functionals of vector valued stationary Gaussian fields. 
Here we provide the foundation needed to carry out that proof.

\section{Introduction. An overview of the results.}

Let $X(p)=(X_1(p),\dots,X_d(p))$, $p\in{\mathbb Z}^\nu$, where 
${\mathbb Z}^\nu$ denotes the lattice points with integer 
coordinates in the $\nu$-dimensional Euclidean space ${\mathbb R}^\nu$, 
be a $d$-dimensional real valued Gaussian stationary random 
field with expectation $EX(p)=0$, $p\in{\mathbb Z}^\nu$. We 
define the notion of Gaussian property of a random field in 
the usual way, i.e. we demand that all finite sets 
$(X(p_1),\dots,X(p_k))$, $p_j\in{\mathbb Z}^\nu$, $1\le j\le k$, 
be a Gaussian random vector, and we call a random field $X(p)$, 
$p\in {\mathbb Z}^\nu$, stationary if for all 
$m\in{\mathbb Z}^\nu$ the random field $X^{(m)}(p)=X(p+m)$, 
$p\in{\mathbb Z}^\nu$, has the same finite dimensional 
distributions as the original random field $X(p)$, 
$p\in{\mathbb Z}^\nu$. In most works only the case $\nu=1$ 
is considered, but since we can prove our results without 
any difficulty for stationary random fields with arbitrary 
parameter $\nu\ge1$ we consider such more general models.

Our goal is to work out a good calculus which provides such a 
representation of the non-linear functionals of our vector
valued Gaussian stationary random field which helps us 
in the study of limit theorems for such functionals. To
understand what kind of limit theorems we have in mind
take the following example which is discussed in Section~8
of this paper.

Let us have a function $H(x_1,\dots,x_d)$ of $d$ variables,
and define with the help of a $d$-dimensional vector valued 
Gaussian stationary random field 
$$
X(p)=(X_1(p),\dots,X_d(p)), \quad p\in{\mathbb Z}^\nu,  
$$
and this function the
random variables $Y(p)=H(X_1(p),\dots,X_d(p))$ for all 
$p\in{\mathbb Z}^\nu$. Let us introduce for all $N=1,2,\dots$
the normalized sum
\begin{equation}
S_N=A_N^{-1}\sum_{p\in B_N} Y(p) \label{1.1} 
\end{equation}
with an appropriate norming constant $A_N>0$, where
$$
B_N=\{p=(p_1,\dots,p_\nu)\colon\; 0\le p_k<N \textrm { for all } 
1\le k\le\nu\}. 
$$
We are interested in a limit theorem for these normalized
sums $S_N$ with an appropriate norming constant $A_N$ as
$N\to\infty$. We are interested in the case when there is
a relatively strong dependence between the elements of the
random field $X(p)$, $p\in{\mathbb Z}^\nu$, and new
kind of limit theorems can appear.

A. M.~Arcones studied a similar problem in paper~\cite{1}. 
He considered only the case $\nu=1$, but this is not 
a serious restriction. The main point is that he 
investigated the non-linear functionals of {\it vector
valued}\/ stationary Gaussian random sequences, and such
problems were not studied before. He was interested both 
in the question when the above defined normalized sums 
$S_N$ satisfy the central limit theorem with the usual 
normalization $A_N=\sqrt N$, and when we get a new
kind of limit theorem with an unorthodox normalization.

First he proved a classical central limit theorem if the 
covariance function $r(p)$, $p\in{\mathbb Z}^1$, of the
underlying vector valued Gaussian stationary random 
sequence tends to zero sufficiently fast at infinity, and 
the function $H(x_1,\dots,x_d)$ satisfies some nice properties. 
This result can be considered the multivariate generalization 
of the result in~\cite{3}, and the proof is based on a refinement 
of the arguments of this paper. Then he presented in Theorem~6 
a non-central limit theorem for $S_N$ which can be considered 
as the multivariate generalization of a non-central limit 
theorem for the random sums $S_N$ proved in~\cite{6} under 
appropriate conditions for a scalar valued Gaussian 
stationary random field, i.e. in the case $d=1$. However, I 
found the proof (and formulation) of this result incomplete.

In the proof of~\cite{6} a crucial point was the representation 
of the random variables $S_N$ in the form of a multiple 
Wiener--It\^o integral with respect to the random spectral 
measure of the underlying stationary random field. With its help 
we could carry out a limiting procedure that enabled us to 
express the limit by means of a multiple Wiener--It\^o integral. 
We did this in the scalar valued case. A detailed explanation
of this approach is given in~\cite{11}. But the results applied in 
this approach were proved only in the scalar valued case.
In the proof of Arcones which is based on similar arguments
one would need the vector valued version of the notions and 
results discussed in~\cite{11}. But they were not available. 

I wanted to replace Arcones' discussion about Theorem~6 of 
his paper with a full proof of this result. It turned out 
that to do this first I have to work out the multivariate 
version of the theory presented in~\cite{11} which dealt with 
the scalar valued case. This is the subject of the present 
paper. It can be considered as a continuation of the research
of It\^o in~\cite{10} and of Dobrushin in~\cite{5}. 

It\^o considered a Gaussian random field in \cite{10} whose 
elements could be expressed as random integrals with respect 
to a Gaussian orthogonal random measure. He defined multiple 
random integrals (called later Wiener--It\^o integrals in the 
literature) with respect to this orthogonal random measure, 
and expressed all square integrable random variables 
measurable with respect to the $\sigma$-algebra 
generated by the elements of the Gaussian random field as 
a sum of such multiple integrals. This notion turned out 
to be useful, because it helped in the study of the non-linear   
functionals of the Gaussian random field. In particular, he
found a very useful relation between the multiple random
integrals he defined and Hermite polynomials. 

Later Dobrushin worked out a version of this theory in~\cite{5},  
where he studied the non-linear functionals of a stationary 
Gaussian random field. In such a case a spectral and a random 
spectral measure can be defined in such a way that the 
elements of the stationary Gaussian random field can be 
expressed as the Fourier transform of the random spectral
measure.  Dobrushin introduced the multiple random integral
with respect to this random spectral measure, and studied
its properties. He proved that this new integral has similar
properties as the original multiple integral introduced by
It\^o. This new integral turned out to be useful, because
it made possible to combine It\^o's theory with the Fourier
analysis. A detailed discussion of these theories can be
found in my work~\cite{11}.

We would like to apply Dobrushin's theory if we are dealing
with vector valued stationary Gaussian random field. But
we cannot do this in a direct way, because Dobrushin's
theory applies some results which are valid in their
original form only for scalar valued random fields. So we 
have to work out the vector valued version of this theory, 
and this is the subject of the present work. 

First we have to find the multivariate version of the
spectral and random spectral measure, and this is the
subject of Sections~2 and~3. Moreover, we need later
the notion of spectral measure and random spectral
measure corresponding to stationary generalized
random fields, and this is the subject of Section~4.
(The precise definition of the notions mentioned in
this introduction will be given in the more detailed
discussion of the main text.)

Then I define the multiple Wiener--It\^o integrals with 
respect to the coordinates of a vector valued random
spectral measure in Section~5. In Section~6 I prove the 
diagram formula which enables us to express the product 
of two multiple Wiener--It\^o integrals as the sum of 
appropriately defined multiple Wiener--It\^o integrals.
This result has an important role in Section~7 which has
two main subjects. The first one is to create a useful 
relation between multiple Wiener--It\^o integrals and the
multivariate generalization of Hermite polynomials, the
so-called Wick polynomials. The second subject of Section~7 
is a useful representation of the shift transformation
of a random variable given in the form of a multiple 
Wiener--It\^o integral. These results enable us to work 
out a method in Section~8 to prove new type non-central 
limit theorems, in particular to give a correct 
formulation and proof of Theorem~6 of Arcones 
paper~\cite{1}. Next I give a more detailed description
about the content of the subsequent sections.

\subsection{A more detailed description of the results.}

In working out the multivariate version of the
theory in~\cite{11} first I characterize the distribution 
of the vector valued Gaussian stationary random fields 
$X(p)=(X_1(p),\dots,X_d(p))$, $p\in{\mathbb Z}^\nu$, 
with expectation zero. This is the subject of the 
second section of this work. Because of the Gaussian 
and stationary property of such a random field its 
distribution is determined by the correlation 
function $r_{j,j'}(p)=EX_j(0)X_{j'}(p)$ for all 
$1\le j,j'\le d$ and $p\in{\mathbb Z}^\nu$. We are 
interested in the description of those functions 
$r_{j,j'}(p)$ which can appear as the correlation 
function of a vector valued stationary random field.

In the scalar valued case a well-known result solves
this problem. The correlation function $r(p)=EX(0)X(p)$,
$p\in{\mathbb Z}^\nu$, of a stationary field $X(p)$, 
$p\in{\mathbb Z}^\nu$, can be represented in a unique
way as the Fourier transform of a spectral measure,
and the spectral measure can be characterized. Namely,
we call the finite (non negative), even measures on the 
torus $[-\pi,\pi)^\nu$ spectral measures. For any 
correlation function $r(p)$ of a stationary field 
there is a unique spectral measure $\mu$ such
that $r(p)=\int e^{i(p,x)}\mu(\,dx)$ for all 
$p\in{\mathbb Z}^\nu$, and for all spectral
measures~$\mu$ there is a (Gaussian) stationary random
field whose correlation function equals the Fourier
transform of this spectral measure~$\mu$.

In Section~2 we prove a similar result for vector 
valued stationary random fields. In the case of a 
vector valued Gaussian stationary random field 
$X(p)=(X_1(p),\dots,X_d(p))$, $p\in{\mathbb Z}^\nu$, 
we have for all pairs of indices $(j,j')$, 
$1\le j,j'\le d$, a unique complex  
measure $G_{j,j'}$ on the torus $[-\pi,\pi)^\nu$ with 
finite total variation such that 
$r_{j,j'}(p)=EX_j(0)X_{j'}(p)=\int e^{i(p,x)}G_{j,j'}(\,dx)$
for all $p\in{\mathbb Z}^\nu$. This can be interpreted
so that the correlation function $r_{j,j'}(p)$, 
$1\le j,j'\le d$, $p\in{\mathbb Z}^\nu$, is the 
Fourier transform of a matrix valued measure 
$(G_{j,j'})$, $1\le j,j'\le d$, on the torus 
$[-\pi,\pi)^\nu$. We want to give, similarly to the
scalar valued case, a complete description of those
matrix valued measures on the torus $[-\pi,\pi)^\nu$ 
for which the correlation function of a vector valued  
Gaussian stationary random field can be represented 
as its Fourier transform. Such matrix valued measures
will be called matrix valued spectral measures.

As I have mentioned, the coordinates of a matrix
valued spectral measure are complex
measures with finite total variation. 
The scalar valued counterpart of this condition
is the condition that the spectral measure of
a scalar valued stationary random field must be 
finite. Another important property of a matrix 
valued spectral measure is that it must be 
positive semidefinite. The meaning of this 
property is explained before the formulation
of Theorem~2.2, and Lemma~2.3 gives a different,
equivalent characterization of this property. 
Let me remark that in the scalar valued case 
the spectral measure must be a measure (and 
not only a complex measure), 
and this fact corresponds to the above property 
of matrix valued spectral measures. Finally, a 
matrix valued spectral measure must be even. 
This means that its coordinates are even, i.e. 
for all $1\le j,j'\le d$ and measurable sets 
$A$ on the torus 
$G_{j,j'}(-A)=\overline{G_{j,j'}(A)}$, where
the overline indicates complex conjugate. 

Theorem~2.2 states that the above properties 
characterize the matrix valued spectral measures.
Let me remark that there are papers, (see e.g. 
\cite{4}, \cite{8} or \cite{15}) which contain the above 
results, although in a slightly different
formulation at least in the case $\nu=1$. 
Nevertheless, I worked out their proof, 
since I applied a different method which 
is used also in the later part of the paper.

In Section 3 I consider the vector valued random 
spectral measures which correspond to a matrix valued 
spectral measure $(G_{j,j'})$, $1\le j,j'\le d$. 
They are sets of vector valued random vectors 
$(Z_{G,1}(A),\dots,Z_{G,d}(A))$ defined for all
measurable subsets $A\subset[-\pi,\pi)^\nu$ on 
the torus with some nice properties which enable 
us to define random integrals with respect to 
them. Let me remark that all random variables
$Z_{G,j}(A)$, $1\le j\le d$, $A\subset[-\pi,\pi)^\nu$, 
are complex valued. The correlation function 
of a vector valued Gaussian random field 
$X(p)=(X_1(p),\dots,X_d(p))$, $p\in{\mathbb Z}^\nu$, 
can be expressed as the Fourier transform of 
its matrix valued spectral measure. One of the 
main results in Section~3 states that a 
random spectral measure $(Z_{G,1},\dots,Z_{G,d})$
can be constructed in such a way that its 
Fourier transform expresses the random field 
itself. More explicitly, 
$X_j(p)=\int e^{i(p,x)}Z_{G,j}(\,dx)$ for all 
$p\in{\mathbb Z}^\nu$ and $1\le j\le d$.
I have listed some properties of this random
measure $(Z_{G,1},\dots,Z_{G,d})$ which determine 
its distribution, and we call a vector valued 
random measure with such properties a vector 
valued random spectral measure corresponding 
to the matrix valued spectral measure 
$(G_{j,j'})$, $1\le j,j'\le d$. We can prove 
that the Fourier transform of a vector 
valued random spectral measure corresponding 
to a matrix valued spectral measure is a 
vector valued Gaussian stationary random field 
with this matrix valued spectral measure.

Besides the result  that for all matrix valued 
spectral measures $(G_{j,j'})$, $1\le j,j'\le d$, 
there exists a vector valued random spectral 
measure corresponding to it I also proved some 
important properties of the random integrals 
with respect to a vector valued spectral 
measure in Section~3. I also proved that if a 
vector valued Gaussian stationary random field 
$X(p)=(X_1(p),\dots,X_d(p))$, $p\in{\mathbb Z}^\nu$, 
is given, we fix some parameter $1\le j\le d$, 
and take the real Hilbert space consisting of 
the closure of finite linear combinations
$\sum_k c_kX_j(p_k)$ with real number valued 
coefficient $c_k$ in the Hilbert space of 
square integrable random variables, then
each element of this Hilbert space can be 
expressed as the integral of a function on
the torus~$[-\pi,\pi)^\nu$ with respect to 
the random spectral measure $Z_{G,j}$. The
functions taking part in the representation
of this Hilbert space also constitute a
real Hilbert space. A more detailed 
formulation of this result is given in 
Lemma~3.2.

It may be worth discussing the relation of 
the results in Section~3 to their scalar 
valued correspondents. The results about
the existence of random spectral measures 
for scalar valued Gaussian stationary 
random fields give a great help in proving 
the results in Section~3. In particular, 
these results provide the definition of the 
random spectral measures $Z_{G,j}$, and 
determine their distribution for all 
$1\le j\le d$. The definition of $Z_{G,j}$, 
and the properties determining its 
distribution depend only on the 
measure~$G_{j,j}$. On the other hand,
we had to carry out some additional work 
to prove those properties of a vector 
valued spectral random measure which 
determine the joint distribution of their 
coordinates. The complex  
measures $G_{j,j'}$ with $j\neq j'$ appear 
at this point of the investigation.

The fourth section deals with a special 
subject, and our motivation to study it 
demands some explanation. Here we consider 
vector valued Gaussian stationary generalized 
random fields.

We could have considered the continuous time version of 
vector valued stationary random fields where the parameter 
set is $t\in {\mathbb R}^\nu$ and not $p\in{\mathbb Z}^\nu$. We 
did not discuss such models, we have considered instead 
vector valued Gaussian stationary generalized random 
fields. This means a set of random vectors  
$(X_1(\varphi),\dots,X_d(\varphi))$ with some nice
properties which are indexed by an appropriately chosen
class of functions. The precise definition of this notion 
is given in Section~4. We have constructed a large 
class of Gaussian stationary generalized random fields, 
presented their matrix valued spectral measures, and 
constructed the vector valued random spectral measures
corresponding to them. In \cite{11} the notion of Gaussian 
stationary generalized random fields was introduced
and investigated in the scalar valued case. Some
useful results were proved there. It was shown (with 
the help of some important results of Laurent Schwartz 
about generalized functions) that in the scalar valued 
case the class of Gaussian, stationary generalized 
random fields constructed in such a way as it was 
done in the present paper contains all Gaussian 
stationary generalized random fields. (Here I 
consider two random fields the same if their finite 
dimensional distributions agree.) This result is probably 
also valid in the case of vector valued generalized random 
fields, but I did not study this question, because 
I was interested in a different problem.

Although the theory of generalized random fields is 
an interesting subject in itself, I was interested 
not so much in their properties, I was investigating 
them for a different reason. I was interested in 
the matrix valued spectral measures of vector
valued Gaussian stationary generalized random
fields and the vector valued random spectral 
measures corresponding to them and not in the
Gaussian, stationary generalized random fields 
which were needed for their construction. They 
behave similarly to the analogous objects
corresponding to (non-generalized) Gaussian
stationary random fields. We can work with them
in the same way. Nevertheless, there is a difference
between these new spectral and random spectral 
measures and the previously defined ones which is 
very important for us. Namely, the coordinates of 
a matrix valued spectral measure corresponding to 
a non-generalized random field are complex  
measures with finite total variation, while 
in the case of generalized random fields the matrix 
valued spectral measures need not satisfy this condition. 
It is enough to demand that the corresponding matrix 
valued measures have locally finite total variation, 
and the matrix valued spectral measures are semidefinite
matrix valued measures with moderately increasing
distribution at infinity. (The definition of these
notions is contained in Section~4.)

The above facts mean that we can work with a much
larger class of random spectral measures after the
introduction of Gaussian stationary generalized random
fields and random spectral measures corresponding
to them. This is important for us, because in the limit
theorems we are interested in the limit can be expressed
by means of multiple Wiener--It\^o integrals with
respect to random spectral measures  constructed with 
the help of vector valued Gaussian stationary 
generalized random fields.

Sections 2---4 contain the main results about the 
linear functionals of vector valued Gaussian
stationary random fields we need in our investigation.
They are also needed in the study of non-linear 
functionals of vector valued stationary Gaussian fields, 
which is the subject of the remaining part of this work. 
Here I work out some tools which are useful in the study 
of limit theorems with a new type of non-Gaussian limit.

In Section 5 multiple Wiener--It\^o integrals are defined
with respect to the coordinates of a vector valued random
spectral measure $(Z_{G,1},\dots,Z_{G,d})$. We define for
all numbers $n=1,2,\dots$, and parameters 
$j_1,\dots,j_n$ such that $1\le j_k\le d$ for all 
$1\le k\le n$ and all functions $f\in{\cal K}_{n,j_1,\dots,j_n}$,
where ${\cal K}_{n,j_1,\dots,j_n}$ is a real Hilbert space 
defined in Section~5 an $n$-fold Wiener--It\^o integral
$$
I_n(f|j_1,\dots,j_n)=
\int f(x_1,\dots,x_n)Z_{G,{j_1}}(\,dx_1)\dots Z_{G,{j_n}}(\,dx_n),
$$
and prove some of its basic properties. The definition and 
proofs are very similar to the definition and proofs in 
scalar valued case, only we have to apply the properties 
of vector valued random spectral measures.

There is one point where we have a weaker estimate than
in the scalar valued case. We can give an upper bound on 
the second moment of a multiple Wiener--It\^o integral 
with the help of the $L_2$ norm of the kernel function 
of this integral in the way as it is formulated in
formula~(\ref{5.6}), but we can state here only an inequality 
and not an equality. The behaviour of Wiener--It\^o 
integrals with respect to a scalar valued random spectral 
measure is different. If we integrate in this case a 
symmetric function, and we may restrict our attention to 
such integrals, then we have equality in the corresponding 
relation. This weaker form of the estimate~(\ref{5.6})  
has the consequence that in certain problems we can get 
only weaker results for Wiener--It\^o integrals with 
respect to the coordinates of a vector valued random 
spectral measure than for Wiener--It\^o integrals with  
respect to scalar valued random spectral measures. But 
as it turns out we can work well with multiple 
Wiener--It\^o integrals with respect to the 
coordinates of a vector valued spectral measure.

The multiple Wiener--It\^o integrals were introduced
in order to express a large class of random 
variables with their help. More precisely, we are 
interested in the following problem. Let us have a 
vector valued Gaussian stationary random field 
$X(p)=(X_1(p),\dots,X_d(p))$, $p\in{\mathbb Z}^\nu$. 
Their elements can be expressed as the  Fourier 
transforms of a vector valued random spectral 
measure $Z_G=(Z_{G,1},\dots,Z_{G,d})$. Let us consider
the real Hilbert space ${\cal H}$ defined in the
second paragraph of Section~5 with the help
of this vector valued stationary Gaussian random field.
We would like to express the elements of this Hilbert 
space in the form of a sum of multiple Wiener--It\^o 
integrals with respect to the coordinates of the 
vector valued spectral measure~$Z_G$.

Let ${\cal H}_1$ denote the linear subspace of ${\cal H}$
generated by the finite linear combinations of the random
variables $X_j(p)$, $p\in{\mathbb Z}^\nu$, $1\le j\le d$,
(with real valued coefficients). It follows from Lemma~3.2
that the random variables $\xi\in{\cal H}_1$ which can 
be written as the sum of one-fold Wiener--It\^o integrals 
$\xi=\sum_{j=1}^d\int f_j(x)Z_{G,j}(\,dx_j)$ with some
functions $f_j\in{\cal K}_{1,j}$, $1\le j\le d$ constitute
an everywhere dense linear subspace in ${\cal H}_1$. Besides,
we can define for all $n=0,1,2,\dots$ some appropriate 
orthogonal subspaces ${\cal H}_n$ of the Hilbert space 
${\cal H}$ such that ${\cal H}$ is the direct sum
${\cal H}={\cal H}_0+{\cal H}_1+{\cal H}_2+\cdots$
of these subspaces. This result is described in Section~7
in more detail. In the case of scalar valued random 
fields it was proved (see~\cite{11}) that all elements of 
${\cal H}_n$ can be expressed as an $n$-fold Wiener--It\^o 
integral. We would like to prove a similar result for
vector valued fields. This is one of the main subjects
of the investigations in Sections~6 and~7.

In the case of vector valued Gaussian stationary random
fields we can prove only a weaker result. We can write
the elements of a dense linear subspace of 
${\cal H}_n$ in the form of a sum $n$-fold Wiener--it\^o 
integrals. This subspace contains all polynomials of 
the random variables $X_j(p)$, $1\le j\le d$, 
$p\in{\mathbb Z}^\nu$. On the other hand, we can write 
the presentation of these elements in a more or less 
explicit form. The reason for being able to do this 
only for the elements of a dense subspace of 
${\cal H}_n$ is related to the weakness of the 
estimate~(\ref{5.6}) mentioned before.

In Section~6 I formulate and prove the multivariate 
version of a classical result. I describe the product 
of two multiple Wiener--It\^o integrals as the sum of 
multiple Wiener--It\^o integrals with respect to the
coordinates of a vector valued random spectral measure.
The formulation and proof of this result is similar
to that of the corresponding result in the scalar 
valued case. In this result we define the kernel 
functions of the Wiener--It\^o integrals appearing 
in the sum expressing the product of two Wiener--It\^o 
integrals with the help of some diagrams. Hence this 
result got the name diagram formula. I wrote down 
the formulation of the diagram formula in the case 
of vector valued random spectral measures in detail. 
On the other hand, I gave only a sketch of proof of 
the diagram formula, because it is actually an 
adaptation of the original proof with a rather 
unpleasant notation. I concentrated on the points 
which explain why the diagram formula has such a 
form as we claim. Besides, I tried to explain those 
steps of the proof where we have to apply some 
new ideas. I hope that the interested reader can 
reconstruct the proof on the basis of these 
explanations by looking at the original proof.

Section 6 also contains a corollary of the diagram
formula, where I formulate this result in a special
case. I formulated this corollary, because in this
work we need only this corollary of the diagram formula.

Section 7 has two main subjects. One of them is
the multivariate version of It\^o's formula which
enables us to rewrite Wick polynomials in the form
of multiple Wiener--It\^o integrals, the other 
one is a useful formula which expresses the shift 
transformations of a random variable given in the 
form of a multiple Wiener--It\^o integral. Although 
the introduction and investigation of the shift
transformations were introduced only in 
Section~7, their investigation is a most 
important subject of the present work. We can 
prove the limit theorems we are interested in 
with the help of the representation of the shift 
transformations given in Section~7.

In the original version of It\^o's formula the
 Hermite polynomial of a standard normal random 
variable or the product of Hermite polynomials 
of independent standard normal random variables 
is expressed in the form of a multiple 
Wiener--It\^o integral. Its multivariate version 
is a similar result about Wick polynomials. Wick 
polynomials are the natural multivariate versions 
of Hermite polynomials of standard Gaussian
random variables.

In Section~7 first I recall from~\cite{11} the 
definition of Wick polynomials together with 
their most important properties. Given 
a homogeneous polynomial of order $n$ of some 
random variables $U_p\in{\cal H}_1$ we define 
the Wick polynomial corresponding to it as it 
is written down in this section. In Theorem~7.2 
and in its corollary we express a Wick polynomial 
of order~$n$ as a sum of  Wiener--It\^o 
integrals of order $n$ if the random variables 
$U_p$ appearing in the definition of this Wick
polynomial are given in the form 
$U_p=\int\varphi_p(x) Z_{G,{j_p}}(\,dx)$ with some
parameter $1\le j_p\le d$ and 
$\varphi_p\in{\cal K}_{1,j_p}$. The proof of the 
original version of It\^o's formula was based
on the diagram formula and the recursive relation 
$H_n(x)=xH_{n-1}(x)-(n-1)H_{n-2}(x)$ for Hermite
polynomials. The multivariate version of 
It\^o's formula is proved with the help 
of the corollary of the diagram formula 
formulated in Section~6 and a 
multivariate version of the above mentioned 
recursive formula for Wick polynomials which is 
proved in Proposition~7.1.

Next I recall the definition of the shift
transformations and show how the shift
transformations of a random variable given in 
the form of a multiple Wiener--It\^o integral 
can be calculated. This result is interesting 
for us, because as the corollary of Theorem~7.2
and Proposition~7.3 indicate, a large class of 
random variables can be written in the form of 
a sum of multiple Wiener--It\^o integrals.

The formula for the calculation of the
shift transform for Wiener--It\^o 
integrals is very similar to the
analogous result about the shift 
transformations of a Wiener--It\^o 
integral with respect to a scalar valued random 
spectral measure. This result, formulated in 
Proposition~7.4, states that if
$$
Y=\int h(x_1,\dots,x_n)Z_{G,{j_1}}(\,dx_1)\dots Z_{G,{j_n}}(\,dx_n),
$$
then its shift transformation $T_u$ equals
$$
T_uY=\int e^{i(u,x_1+\cdots+x_n)}
 h(x_1,\dots,x_n)Z_{G,{j_1}}(\,dx_1)\dots Z_{G,{j_n}}(\,dx_n). 
$$
for all $u\in{\mathbb Z}^\nu$.

In Section 8 a method for proving limit theorems for 
non-linear functionals of vector valued Gaussian 
stationary fields is discussed. The example mentioned
at the start of this Introduction  is considered. Some 
standard arguments show that in this problem we can 
restrict our attention to the case when the random 
variables $Y(p)=H(X_1(p),\dots,X_d(p))$ in the 
normalized sum we are investigating are Wick polynomials.

Thus we consider in Section~8 the above limit problem in the
special case when  $Y(p)=H(X_1(p),\dots,X_d(p))$ is a Wick 
polynomial (for all parameters $p\in{\mathbb Z}^\nu$). In 
this case we can rewrite the normalized sums $S_N$ in a 
useful form with the help of the multivariate version of 
It\^o's formula and our result about the representation 
of the shift transformation of multiple Wiener--It\^o
integrals. We rewrite these normalized sums in such a
form that provides a heuristic argument for the proof
of a limit theorem.

Our goal is to find a result that enables us to make
a precise proof on the basis of this heuristic argument.
The main result of Section~8, Proposition 8.1 may help
us to carry out such a program. Besides, we prove another
result in Lemma~8.2 that simplifies a little bit the 
conditions of Proposition~8.1.

In a subsequent paper~\cite{13} I shall prove a 
multivariate version of the limit theorem in~\cite{6} 
with the help of the present paper. In that proof I shall
apply Proposition~8.1, and the main problem is to check 
its conditions. The result in~\cite{6} was proved with 
the help of the scalar valued version of the results in 
this work. In that proof it had to be shown that the 
conditions of the scalar valued version of 
Proposition~8.1 hold under appropriate assumptions. 
We did it by proving a relation between the
behaviour of the correlation function and spectral 
measure of a stationary random  field. In the proof 
of the multivariate generalization of the result in 
paper~\cite{6} it will be shown that a similar 
relation holds also between the behaviour of the 
correlation function and spectral measure of a vector 
valued stationary random field.

\section{Spectral representation of vector valued 
stationary random fields}

Let $X(p)=(X_1(p),\dots,X_d(p))$, $p\in{\mathbb Z}^\nu$, where 
${\mathbb Z}^\nu$ denotes the lattice of points with integer 
coordinates in the $\nu$-dimensional Euclidean space 
${\mathbb R}^\nu$, be a $d$-dimensional real valued Gaussian 
stationary random field with expected value $EX(p)=0$, 
$p\in{\mathbb Z}^\nu$. Let us first characterize the 
covariance matrices $R(p)=(r_{j,j'}(p))$, $1\le j,j'\le d$, 
$p\in{\mathbb Z}^\nu$ of this $d$-dimensional stationary 
random field, where
$r_{j,j'}(p)=EX_j(0)X_{j'}(p)=EX_j(m)X_{j'}(p+m)$, 
$1\le j,j'\le d$, $p,m\in{\mathbb Z}^\nu$.

In the case $d=1$ we can characterize the function $R(p)=EX(0)X(p)$,
(in this case $j=j'=1$, so we can omit these indices) as the Fourier
transform of an even finite (and positive) measure $G$ on the torus 
$[-\pi,\pi)^\nu$, called the spectral measure. We are looking for 
the vector valued version of this result. Before discussing this
problem I recall the definition of the torus $[-\pi,\pi)^\nu$. 

The points of the torus $[-\pi,\pi)^\nu$ are those points 
$x=(x_1,\dots,x_\nu)\in {\mathbb R}^\nu$ for which 
$-\pi\le x_j\le\pi$ for all $1\le j\le\nu$. But if a coordinate 
of $x$ in this set equals $\pi$, then we consider this point 
the same if we replace this coordinate by~$-\pi$. In such 
a way we can identify all points of this set by a point of 
the set $[-\pi,\pi)^\nu\subset {\mathbb R}^\nu$. We define 
the topology on the torus on $[-\pi,\pi)^\nu$ as the topology 
induced by the metric 
$\rho(x,y)=\sum\limits_{j=1}^\nu(|x_j-y_j|\;\textrm{mod}\;2\pi)$
if $x=(x_1,\dots,x_\nu)\in[-\pi,\pi)^\nu$ and 
$y=(y_1,\dots,y_\nu)\in[-\pi,\pi)^\nu$. These properties of the 
torus $[-\pi,\pi)^\nu$ must be taken into account when we 
speak of the set $-A=\{-x\colon\; x\in A\}$ for a set 
$A\subset [-\pi,\pi)^\nu$ or of a continuous function on the 
torus $[-\pi,\pi)^\nu$.

Later we shall speak also about the torus $[-A,A)^\nu$ for arbitrary
$A>0$. This is defined in the same way, only the number $\pi$ is 
replaced by~$A$ in the definition.

%\medskip\noindent
%{\it Remark:}\/ In the literature the $\nu$-dimensional torus
%is defined generally as the factor space 
%${\mathbb T}^\nu={\mathbb R}^\nu/(2\pi{\mathbb Z})^\nu$. 
%Nevertheless, I chose the above equivalent definition.
%I chose this definition, because in our further investigation
%it will be important to consider the convergence of (rescaled
%versions) of complex measures on the torus to a complex measure 
%on ${\mathbb R}^\nu$. (See the vague convergence introduced and
%investigated in Section~8.) In this
%investigation we have to work with measures on the Euclidean
%space, so it is better if they are defined there from the start.

\medskip\noindent

It is natural to expect that there is a natural definition of 
even positive semidefinite matrix valued measures also in the 
$d$-dimensional case, $d\ge2$, and this takes the role of the 
spectral measure in the vector valued case. To define this 
notion first I prove a lemma. Before formulating it I recall 
the definition of a complex measure with finite 
total variation, since this notion appears in the formulation 
of the lemma. We say that a complex measure on
a measurable space has finite total variation if both its real
and imaginary part can be represented as the difference of 
two finite measures. I also recall Bochner's theorem, more 
precisely that version of this result that we shall apply in 
the proof.

\medskip\noindent
{\bf Bochner's theorem.} {\it Let $f(p)$, $p\in{\mathbb Z}^\nu$,
be a positive definite function on ${\mathbb Z}^\nu$, i.e. such 
a function for which the inequality
$\sum\limits_{j=1}^N\sum\limits_{j'=1}^N z_j\bar z_{j'}f(p_j-p_{j'})\ge0$
holds for any set of points $p_j\in{\mathbb Z}^\nu$, and complex 
numbers $z_j$, $1\le j\le N$, with some number $N\ge1$. Then 
there exists a unique finite measure $G$ on the torus 
$[-\pi,\pi)^\nu$ such that
$$
f(p)=\int_{[-\pi,\pi)^\nu}e^{i(p,x)}G(\,dx) \quad \textrm{for all }
p\in{\mathbb Z}^\nu.
$$
If the function $f$ is real valued, then the measure $G$ is
even, i.e. $G(-A)=G(A)$ for all $A\subset[-\pi,\pi)^\nu$.}

\medskip
Next I formulate the following lemma.

\medskip\noindent
{\bf Lemma 2.1.} {\it Let $X(p)=(X_1(p),\dots,X_d(p))$, 
$p\in{\mathbb Z}^\nu$, be a $d$-dimensional stationary 
Gaussian random field with expectation zero. Then for 
all pairs $1\le j,j'\le d$ the correlation function 
$r_{j,j'}(p)=EX_j(0)X_{j'}(p)$, $p\in{\mathbb Z}^\nu$, 
can be written in the form
\begin{equation}
r_{j,j'}(p)=EX_j(0)X_{j'}(p)=EX_j(m)X_{j'}(m+p)
=\int_{[-\pi,\pi)^\nu} e^{i(p,x)}G_{j,j'}(\,dx) \label{2.1}
\end{equation}
with a complex measure $G_{j,j'}$ on the torus 
$[-\pi,\pi)^\nu$ with finite total variation. The function 
$r_{j,j'}(p)$, $p\in{\mathbb Z}^\nu$, uniquely determines this 
complex measure $G_{j,j'}$ with finite total 
variation. It is even, i.e. $G_{j,j'}(-A)=\overline{G_{j,j'}(A)}$ 
for all measurable sets $A\subset[-\pi,\pi)^\nu$. The relation 
$G_{j',j}(A)=\overline{G_{j,j'}(A)}$ also holds for all 
$1\le j,j'\le d$ and $A\subset[-\pi,\pi)^\nu$.}

\medskip\noindent
{\it Remark.}\/ Let us remark that given a $d$-dimensional 
stationary random field with expectation zero, there exist
also such $d$-dimensional stationary random fields with
expectation zero which are Gaussian, and have the same 
correlation function. As a consequence, in Lemma~2.1 we 
could drop the condition that the stationary random field
we are considering is Gaussian. The same can be said about
the other results of Section~2. I imposed this condition,
because later, as we work with random spectral measures and
random integrals with respect to them the Gaussian property
of the underlying random field is important.

\medskip\noindent
{\it Proof of Lemma 2.1.}
By Bochner's theorem we may write
$$
r_{j,j}(p)=\int_{[-\pi,\pi)^\nu} e^{i(p,x)}G_{j,j}(\,dx), \quad 
p\in{\mathbb Z}^\nu, 
$$ 
for all $1\le j\le d$ with some finite measure $G_{j,j}$ on 
$[-\pi,\pi)^\nu$. We find a good representation for $r_{j,j'}(n)$ 
if $j\neq j'$ with the help of following argument. 

The function 
\begin{eqnarray*}
q_{j,j'}(p)&=&E[X_j(0)+iX_{j'}(0)][X_j(p)-iX_{j'}(p)] \\
&=&E[X_j(0)+iX_{j'}(0)]\overline{[X_j(p)+iX_{j'}(p)]},
\end{eqnarray*}
$p\in{\mathbb Z}^\nu$, is positive definite, hence it can be 
written in the form 
$$
E[X_j(0)+iX_{j'}(0)][X_j(p)-iX_{j'}(p)]=\int_{[-\pi,\pi)^\nu}
e^{i(p,x)}H_{j,j'}(\,dx)
$$ 
with some finite measure $H_{j,j'}$ on $[-\pi,\pi)^\nu$.
Similarly,
$$
E[X_j(0)+X_{j'}(0)][X_j(p)+X_{j'}(p)]=\int_{-[\pi,\pi)^\nu} e^{i(p,x)}K_{j,j'}(\,dx)
$$ 
with some finite measure $K_{j,j'}$ on $[-\pi,\pi)^\nu$. Hence
\begin{eqnarray*}
EX_j(0)X_{j'}(p)&=&
\frac i2E[X_j(0)+iX_{j'}(0)][X_j(p)-iX_{j'}(p)] \\
&&\quad +\frac12E[X_j(0)+X_{j'}(0)][X_j(p)+X_{j'}(p)]\\
&&\quad\;-\frac{(1+i)}2 [EX_j(0)X_j(p)+EX_{j'}(0)X_{j'}(p)] \\
&=&\int_{[-\pi,\pi)^\nu} e^{i(p,x)}G_{j,j'}(\,dx)
\end{eqnarray*}
with $G_{j,j'}(\,dx)=\frac12[i H_{j,j'}(\,dx)+K_{j,j'}(\,dx)]
-\frac{(1+i)}2[G_{j,j}(\,dx)+G_{j',j'}(\,dx)]$.

In such a way we have found complex measures 
$G_{j,j'}$ with finite total variation which satisfy relation~(\ref{2.1}). 
Since this relation holds for all $p\in{\mathbb Z}^\nu$, the 
function $r_{j,j'}(p)$, $p\in{\mathbb Z}^\nu$, determines the 
measure $G_{j,j'}$ uniquely.

Since $r_{j,j'}(p)$ is real valued, i.e. 
$r_{j,j'}(p)=\overline{r_{j,j'}(p)}$, it can be written both in 
the form 
$$
r_{j,j'}(p)=\int_{[-\pi,\pi)^\nu} e^{i(p,x)}G_{j,j'}(\,dx)
$$
and 
$$
r_{j,j'}(p)=\int_{[-\pi,\pi)^\nu} e^{-i(p,x)}\overline{G_{j,j'}(\,dx)}=
\int_{[-\pi,\pi)^\nu} e^{i(p,x)}\overline{G_{j,j'}(-\,dx)}.
$$ 
Comparing 
these relations we get that $G_{j,j'}(A)=\overline{G_{j,j'}(-A)}$ 
for all measurable sets $A\subset[-\pi,\pi)^\nu$. Similarly, 
the relation $r_{j',j}(p)=r_{j,j'}(-p)$ implies that 
$G_{j',j}(A)=G_{j,j'}(-A)=\overline{G_{j,j'}(A)}$ for all measurable 
sets $A\subset[-\pi,\pi)^\nu$. Lemma~2.1 is proved.  

\medskip
Since all complex measures $G_{j,j'}$, 
$1\le j,j'\le d$, have finite total variation by Lemma~2.1 
there is a finite measure $\mu$ on the torus $[-\pi,\pi)^\nu$ 
such that all these complex measures $G_{j,j'}$
are absolutely continuous with respect to $\mu$, and the absolute 
value of the Radon--Nikodym derivatives 
$g_{j,j'}(x)=\frac{dG_{j,j'}}{d\mu}(x)$ is integrable with respect
to $\mu$. The properties of the measures $G_{j,j'}$ proved in
Lemma~2.1 imply that the $d\times d$ matrix $(g_{j,j'}(x))$, 
$1\le j,j'\le d$, is Hermitian for almost all 
$x\in [-\pi,\pi)^\nu$ with respect to the measure $\mu$. We 
shall call the matrix valued measure 
$(G_{j,j'}(A))$, $A\subset[-\pi,\pi)^\nu$, positive semidefinite 
if the matrix $(g_{j,j'}(x))$, $1\le j,j'\le d$, is positive 
semidefinite for almost all $x\in [-\pi,\pi)^\nu$ with respect
to $\mu$. More precisely, we introduce the following definition.

\medskip\noindent
{\bf Definition of positive semidefinite matrix valued, even 
measures on the torus.} {\it Let us have some complex  
measures $G_{j,j'}$, $1\le j,j'\le d$, with finite total 
variation on the $\sigma$-algebra of the Borel measurable sets of 
the torus $[-\pi,\pi)^\nu$. Let us consider the matrix valued measure 
$(G_{j,j'})$, $1\le j,j'\le d$. We call this matrix valued measure 
positive semidefinite if there exists a (finite) positive 
measure $\mu$ on $[-\pi,\pi)^\nu$  such that all complex  
measures $G_{j,j'}$, $1\le j,j'\le d$, are absolutely 
continuous with respect to it, and their Radon--Nikodym derivatives 
$g_{j,j'}(x)=\frac{dG_{j,j'}}{d\mu}(x)$, $1\le j,j'\le d$, constitute 
a positive semidefinite matrix $(g_{j,j'}(x))$, $1\le j,j'\le d$
for almost all $x\in{\mathbb Z}^\nu$ with respect to the 
measure~$\mu$. We call this positive semidefinite matrix valued 
measure $(G_{j,j'})$, $1\le j,j'\le d$, on the torus even if 
$G_{j,j'}(-A)=\overline{G_{j,j'}(A)}$ for all measurable sets 
$A\subset[-\pi,\pi)^\nu$ and $1\le j,j'\le d$. 

Later we shall speak also of  positive semidefinite matrix 
valued, even measures on a torus $[-A,A)^\nu$ for arbitrary 
$A>0$ which is defined in the same way, only the complex 
measures $G_{j,j'}$ and the dominating measure 
$\mu$ are defined on $[-A,A)^\nu$.}

\medskip\noindent
{\it Remark.}\/ Here I am speaking about measures with finite
total variation, although such (complex) measures are called
generally bounded measures in the literature. 
Actually, we know by Stone's theorem that any bounded signed 
measure can be represented as the difference of two bounded 
measures (with disjoint support). Nevertheless, I shall remain 
at this name, because actually we prove directly the finite 
total variation of the measures we shall work with in this paper. 
Besides, (in Section~4) I shall
define complex measures on ${\mathbb R}^\nu$ with locally 
finite total variation, and I prefer such a name which refers to
the similarity of these objects. (The complex measures with locally
finite total variation are not measures in the original meaning
of this word, only their restrictions to compact sets are complex 
measures.)

\medskip
The next theorem about the characterization of the correlation 
function of a $d$-di\-men\-sio\-nal stationary Gaussian random 
field with zero expectation states that the correlation 
functions $r_{j,j'}(p)$, $1\le j,j'\le d$, $p\in{\mathbb Z}^\nu$, 
can be given in the form~(\ref{2.1}) with the help of a positive 
semidefinite matrix valued, even measure $(G_{j,j'})$, 
$1\le j,j'\le d$, on the torus $[-\pi,\pi)^\nu$. Moreover,
it will be shown that we have somewhat more freedom in the 
definition of positive semidefinite matrix valued measures 
on the torus. If the coordinates of a matrix valued measure 
$(G_{j,j'})$, $1\le j,k\le d$, are complex  
measures with finite total variation, and this matrix valued 
measure satisfies the definition of the positive semidefinite 
property with some measure~$\mu$, then this measure $\mu$ 
can be replaced in the definition by any such finite 
measure on the torus with respect to which the complex 
measures $G_{j,j'}$ are absolutely continuous. 
More explicitly, the following result holds. 

\medskip\noindent
{\bf Theorem 2.2.} {\it The covariance matrices of a $d$-dimensional
stationary random field $X(p)=(X_1(p),\dots,X_d(p))$, 
$p\in{\mathbb Z}^\nu$,  with expectation zero can be given 
in the following form. For all $1\le j,j'\le d$ there exists 
a complex measure $G_{j,j'}$ with finite total 
variation on the $\nu$-dimensional torus $[-\pi,\pi)^{\nu}$ 
in such a way that for all  $1\le j,j'\le d$ the correlation 
function $r_{j,j'}(p)=EX_j(0)X_{j'}(p)$, $p\in{\mathbb Z}^\nu$, 
is given by formula~(\ref{2.1}) with this complex 
measure~$G_{j,j'}$. The $d\times d$ matrix $G=(G_{j,j'})$, 
$1\le j,j'\le d$, whose coordinates are the complex  
measures $G_{j,j'}$ has the following properties. This 
matrix is Hermitian, i.e. the measures $G_{j,j'}$ satisfy the 
relation $G_{j',j}(A)=\overline{G_{j,j'}(A)}$ for all pairs of indices 
$1\le j,j'\le d$ and measurable sets $A\subset [-\pi,\pi)^\nu$,
and the measures $G_{j,j'}$ are even, i.e. 
$G_{j,j'}(-A)=\overline{G_{j,j'}(-A)}$ for all $1\le j,j'\le d$ and 
$A\subset[-\pi,\pi)^\nu$. For all pairs $(j,j')$, $1\le j,j'\le d$, 
the function $r_{j,j'}(p)$, $p\in{\mathbb Z}^\nu$, defined by 
formula~(\ref{2.1}) uniquely determines the complex  
measure $G_{j,j'}$ with finite total variation. Besides, $G_{j,j'}$
has the following property.

Let us take a finite measure $\mu$ on the torus $[-\pi,\pi)^\nu$ 
such that all complex measures $G_{j,j'}$ are absolutely 
continuous with respect to it, (because of the finite total 
variation of the complex measures $G_{j,j'}$ there 
exist such measures), and put 
$g_{j,j'}(x)=g_{j,j',\mu}(x)=\frac{dG_{j,j'}}{d\mu}(x)$. Then the 
matrix $(g_{j,j'}(x))$, $1\le j,j'\le d$, is positive 
semidefinite for almost all $x\in[-\pi,\pi)^\nu$ with respect
to the measure~$\mu$.

Conversely, if a class of complex measures 
$G_{j,j'}$ on $[-\pi,\pi)^\nu$, $1\le j,j'\le d$, have finite 
total variation, and $(G_{j,j'})$, $1\le j,j'\le d$, is a positive 
semidefinite matrix valued, even measure on the torus, then 
there exists a $d$-dimensional stationary Gaussian field 
$X(p)=(X_1(p),\dots,X_d(p))$, $p\in{\mathbb Z}^\nu$, with 
expectation $EX_j(p)=0$ and covariance 
$EX_j(p)X_{j'}(q)=r_{j,j'}(p-q)$, where the function $r_{j,j'}(p)$ 
is defined in~(\ref{2.1}) with the complex  
measure $G_{j,j'}$ for all parameters $1\le j,j'\le d$ and 
$p,q\in{\mathbb Z}^\nu$.}

\medskip\noindent
{\it Remark.} We shall call the positive semidefinite matrix
valued, even measure $(G_{j,j'})$, $1\le j,j'\le d$, on the torus 
$[-\pi,\pi)^\nu$ with coordinates $G_{j,j'}$ satisfying 
relation~(\ref{2.1}) the matrix valued spectral measure of the 
correlation function $r_{j,j'}(p)$, $1\le j,j'\le d$, 
$p\in{\mathbb Z}^\nu$. In general, we shall call an
arbitrary positive semidefinite matrix valued, even measure 
on the torus $[-\pi,\pi)^\nu$ a matrix valued spectral
measure on the torus $[-\pi,\pi)^\nu$. (More generally, later 
we shall call for any $A>0$ a positive semidefinite matrix 
valued, even measure on the torus $[-A,A)^\nu$ a matrix 
valued spectral measure on this torus.) We have the 
right for such a terminology, since by Theorem~2.2 for an 
arbitrary positive semidefinite matrix valued, even measure 
on the torus $[-\pi,\pi)^\nu$ there is a vector valued 
stationary Gaussian random field on ${\mathbb Z}^\nu$ such 
that this positive semidefinite matrix valued, even measure 
is the spectral measure of its correlation function.

\medskip\noindent
{\it Proof of Theorem 2.2.} The statements formulated in the
first paragraph of Theorem~2.2 follow from Lemma~2.1.
Next we prove that the matrix $(g_{j,j'}(x))$, $1\le j,j'\le d$, 
whose elements are defined as the Radon--Nikodym
derivatives of the complex measures $G_{j,j'}$ 
with respect to a measure $\mu$ satisfying the conditions 
of Theorem~2.2 is positive semidefinite for $\mu$~almost all~$x$.

We prove this by first showing with the help of Weierstrass' 
second approximation theorem that
\begin{equation}
\int_{[-\pi,\pi)^\nu} v(x) g(x) v^*(x)\mu(\,dx)\ge0 \label{2.2}
\end{equation}
for any continuous, $d$-dimensional vector-valued function
\hfill\break 
$v(x)=(v_1(x),\dots,v_d(x))$ on the $\nu$-dimensional torus 
$[-\pi,\pi)^\nu$, where $g(x)$ denotes the $d\times d$ matrix 
$(g_{j,j'}(x))$, $1\le j,j'\le d$, and $v^*(x)$ is the conjugate 
of the vector $v(x)$. 

To prove (\ref{2.2}) let us first observe that by Weierstrass' second 
approximation theorem for all $\varepsilon>0$ there exists a 
number $N=N(\varepsilon)$ and $d$ trigonometrical polynomials 
of order~$N$ 
$$
v_{N,j}(x)=\sum_{\substack{s=(s_1,\dots,s_\nu)\\ 
-N\le s_k< N,\;1\le k\le\nu}}  
a_{j,s_1,\dots,s_\nu}e^{i(s,x)}, \quad 1\le j\le d, \quad
x\in[-\pi,\pi)^\nu
$$
for which 
$$
\sup _{x\in[-\pi,\pi)^\nu}|v_{N,j}(x)-v_j(x)|\le\varepsilon 
\quad\textrm{for all }1\le j\le d.
$$
Let us also define the random vector $Y_N=(Y_{N,1},\dots,Y_{N,d})$ with
coordinates
$$
Y_{N,j}=\sum_{\substack{s=(s_1,\dots,s_\nu)\\ 
-N\le s_k< N, \;1\le k\le\nu}} 
a_{j,s_1,\dots,s_\nu}X_j(s), \quad 1\le j\le d,
$$
Then we have because of the relation 
$EX_j(s)X_{j'}(s')=\int e^{i(s-s',x)}g_{j,j'}(x)\mu(\,dx)$
$$
0\le E\left(\sum_{j=1}^d Y_{N,j}\right)
\overline{\left(\sum_{j=1}^d Y_{N,j}\right)}
=\sum_{j=1}^d\sum_{j'=1}^d \int_{[-\pi,\pi)^\nu}
g_{j,j'}(x)v_{N,j}(x)\overline{v_{N,j'}(x)}\mu(\,dx).
$$
Hence
$$
\int_{[-\pi,\pi)^\nu} v_N(x) g(x) v_N^*(x)\mu(\,dx)\ge0,
$$
and we get relation~(\ref{2.2}) from it with the  help of the limiting 
procedure $N\to\infty$.

Let us choose a vector $a=(a_1,\dots,a_d)\in {\mathbb R}^d$ and a non-negative
continuous function $u(x)$ on the torus $[-\pi,\pi)^\nu$. Let us apply
formula (\ref{2.2}) with the choice of the function 
$v(x)=(a_1\sqrt{u(x)},\dots,a_d\sqrt{u(x)})$. With this choice 
formula (\ref{2.2}) yields that 
$$
0\le\int_{[-\pi,\pi)^\nu} v(x) g(x) v^*(x)\mu(\,dx)
=\int_{[-\pi,\pi)} u(x) h_a(x)\mu(\,dx)
$$
with the function $h_a(x)=a g(x)a^*$. Since this inequality 
holds for all non-negative continuous functions this implies 
that $h_a(x)\ge0$ for almost all $x$ with respect to the 
measure~$\mu$. Moreover, since $h_a(x)=a g(x)a^*$ is a 
continuous function of the parameter~$a$ for a fixed number 
$x\in[-\pi,\pi)^\nu$ this also implies that $g(x)$ is a 
positive semidefinite matrix for almost all $x$ with respect
to the measure~$\mu$. We have proved that the covariance 
matrix of a vector valued stationary field has the properties 
stated in Theorem~2.2.

Next I show that if we have a class of complex 
measures $G_{j,j'}$ with finite total variation such that $(G_{j,j'})$ 
is a positive semidefinite matrix valued even measure on the 
torus, and the functions $r_{j,j'}(p)$, $p\in{\mathbb Z}^\nu$, 
are defined by formula~(\ref{2.1}) with these complex  
measures~$G_{j,j'}$, then there exists a vector-valued stationary 
Gaussian field $X(p)=(X_1(p),\dots,X_d(p))$ with expectation 
zero and covariance function $EX_j(0)X_{j'}(p)=r_{j,j'}(p)$. 

First I show that for all $N\ge1$ there is a set of Gaussian 
random vectors $X(p)=(X_1(p),\dots,X_d(p))$, with parameters 
$p=(p_1,\dots,p_\nu)$, $-N\le p_j\le N$ for all $j=1,\dots,d$, such 
that $EX_j(p)X_{j'}(q)=r_{j,j'}(p-q))$ for all $p=(p_1,\dots,p_\nu)$, 
$q=(q_1,\dots,q_\nu)$ with $-N\le p_s,q_s\le N$, $1\le s\le\nu$. 

Let us observe that the covariances $r_{j,j'}(p)$ defined by~(\ref{2.1}) 
are real-valued, since $G_{j,j'}(A)=\overline{G_{j,j'}(-A)}$. To 
show that there exists a set of Gaussian random vectors with the 
desired covariance we have to check that the covariance matrix 
determined by the coordinates of these random vectors is positive 
semidefinite. This means that for all sets of complex numbers 
$$
{\cal A}_N=\{a_{j,p}=a_{j,p_1,\dots,p_\nu}\colon\;
1\le j\le d, \; -N\le p_s\le N, \textrm{ for all } 1\le s\le\nu\}
$$
$$
I({\cal A}_N)=\sum_{j=1}^d\sum_{j'=1}^d \sum_{\substack{p=(p_1,\dots,p_\nu)\\
-N\le p_s\le N,\; 1\le s\le \nu }}
\sum_{\substack{q=(q_1,\dots,q_\nu)\\
-N\le q_s\le N,\; 1\le s\le \nu}}
a_{j,p}\overline{a_{j',q}} r_{j,j'}(p-q)\ge0.
$$
This inequality holds, since
\begin{eqnarray*}
I({\cal A}_N)&=&\int \sum_{j=1}^d\sum_{j'=1}^d 
\left(
\sum_{\substack{p=(p_1,\dots,p_\nu)\\
-N\le p_s\le N,\; 1\le s\le \nu}} 
a_{j,p} e^{i(p,x)}\right) g_{j,j'}(x) \\
&&\qquad\qquad  
\overline{\left(
\sum_{\substack{p=(p_1,\dots,p_\nu)\\
-N\le p_s\le N,\; 1\le s\le \nu}} 
a_{j',p} e^{i(p,x)}\right)}\mu(\,dx)\\
&&\qquad\qquad\qquad =\int\left( \sum_{j=1}^d\sum_{j'=1}^d 
b_j(x)g_{j,j'}(x)\overline{b_{j'}(x)}\right)\mu(\,dx)\ge0,
\end{eqnarray*}
where $b_j(x)=\sum\limits_{\substack{p=(p_1,\dots,p_\nu)\\
-N\le p_s\le N,\; 1\le s\le \nu}} a_{j,p} e^{i(p,x)}$. This
expression is really non-negative, since the matrix $g_{j,j'}(x)$
is positive semidefinite for $\mu$-almost all $x$, and this implies
that the integrand at the right-hand side of this expression is 
non-negative for $\mu$-almost all~$x$.

Since the distribution of the above sets of Gaussian random vectors
are consistent for different parameters~$N$ it follows from 
Kolmogorov's existence theorem for random processes with consistent 
finite distributions that there exists a Gaussian random field $X(p)$, 
$p\in{\mathbb Z}^\nu$, with $EZ_p=0$, $EX_j(p)X_{j'}(q)=r_{j,j'}(p-q)$,
where $r_{j,j'}(p)$ is defined by formula~(\ref{2.1}) with our matrix 
valued spectral measure $G=(G_{j,j'})$, $1\le j,j'\le d$. In such 
a way we constructed a stationary Gaussian random field with the 
desired properties. Theorem~2.2 is proved.   

\medskip
In the next lemma I give a different characterization of 
positive semidefinite matrix valued, even measures on the torus 
$[-\pi,\pi)^\nu$. 

\medskip\noindent
{\bf Lemma 2.3.} {\it Let us have a class of complex 
measures $G_{j,j'}$, $1\le j,j'\le d$, with finite total variation 
on the torus $[-\pi,\pi)^\nu$.  Let us define with their help the 
following $\sigma$-additive matrix valued function on the  
measurable subsets of the torus $[-\pi,\pi)^\nu$. Define for all 
measurable sets $A\subset[-\pi,\pi)^\nu$ the $d\times d$ matrix 
$G(A)=(G_{j,j'}(A))$, $1\le j,j'\le d$. This matrix valued function
is a positive semidefinite matrix valued, even measure on the torus 
$[-\pi,\pi)^\nu$ if and only if the matrix $(G_{j,j'}(A))$, 
$1\le j,j'\le d$, is positive semidefinite, and 
$G_{j,j'}(-A)=\overline{G_{j,j'}(A)}$ for all measurable sets 
$A\subset[-\pi,\pi)^\nu$ and $1\le j,j'\le d$.}

\medskip\noindent
{\it Proof of Lemma 2.3.}\/ It is clear that if $(G_{j,j'})$ is
a positive semidefinite matrix valued, even measure, then the 
matrix $(G_{j,j'}(A))$ with 
$$
G_{j,j'}(A)=\int_A g_{j,j'}(x)\mu(\,dx), \quad 1\le j,j'\le d,
$$ 
is a positive semidefinite matrix, and 
$G_{j,j'}(-A)=\overline{G_{j,j'}(A)}$ for all measurable sets
$A\subset[-\pi,\pi)^\nu$ and $1\le j,j'\le d$.

On the other hand, it is not difficult to see that if the
above properties hold, then 
$\sum\limits_{j=1}^d\sum\limits_{j'=1}^d
\int v_j(x)\overline{v_{j'}(x)} G_{j,j'}(\,dx)\ge0$
 for all vectors $v(x)=(v_1(x),\dots,v_d(x))$, where $
v_j(\cdot)$, $1\le j\le d$, is a continuous function on the 
torus $[-\pi,\pi)^\nu$. If $\mu$ is a finite measure on 
$[-\pi,\pi)^\nu$ such that  all complex measures 
$G_{j,j'}$, $1\le j,j'\le d$, are absolutely continuous with 
respect to it with Radon--Nikodym derivative $g_{j,j'}(x)$, 
and we denote the matrix $(g_{j,j'}(x))$, $1\le j,j'\le d$, 
by $g(x)$, then the above inequality can be rewritten in the 
form $\int v(x) g(x)v^*(x)\mu(\,dx)\ge0$. In the proof of 
Theorem~2.2 we have seen that this implies that $g(x)$ is
a positive semidefinite matrix for $\mu$ almost all 
$x\in[-\pi,\pi)^\nu$. Lemma~2.3 is proved.

\medskip
Let me also remark that the proof of Lemma~2.3 also implies 
that if the definition of positive semidefinite matrix valued, 
even measures holds with some finite measure $\mu$ on the
torus with the property each the complex  measure $G_{j,j'}$, 
$1\le j,j'\le d$, is absolutely continuous with respect to 
it, then the conditions of this definition also hold  with 
any measure $\mu$ on the torus with the same properties.

Given a positive semidefinite matrix valued even measure 
$G=(G_{j,j'})$, $1\le j,j'\le d$, on the torus $[-\pi,\pi)^\nu$, 
there is a natural candidate for the choice of the measure $\mu$
on the torus $[-\pi,\pi)^\nu$ with respect to which all measures 
$G_{j,j'}$, $1\le j,j'\le d$, are absolute continuous. We shall prove
an estimate in formula~(\ref{3.2}) which implies that the measure 
$\mu=\sum_{j=1}^d G_{j,j}$, i.e. the trace of the matrix valued measure
$G$ has this property. Later this measure will be our choice for the
measure $\mu$.

Let me remark that the proof of Lemma~2.3 yields another characterization
of positive semidefinite matrix valued measures on the torus. I
present it, although I shall not use it later.

A matrix valued measure $G=(G_{j,j'})$, $1\le j,j'\le d$, on the torus
such that $G_{j,j}(A)=\overline{G_{j',j}(A)}$ for all $1\le j,j'\le d$
and measurable sets $A\subset [-\pi,\pi)^\nu$ is positive semidefinite
if and only if
$$
\sum_{j=1}^d\sum_{j'=1}^d \int_{[-\pi,\pi)^\nu} u_j(x)\overline{u_{j'}(x)} 
G_{j,j'}(\,dx)\ge0
$$
for all vectors $u(x)=(u_1(x),\dots,u_d(x))$ whose coordinates are
continuous functions on the torus $[-\pi,\pi)^\nu$.


\section{Random spectral measures in the multi-di\-men\-sio\-nal case}

If $X(p)=(X_1(p),\dots,X_d(p))$, $p\in{\mathbb Z}^\nu$, is a 
$d$-dimensional stationary Gaussian  random field with 
expectation zero, then its distribution is determined by its 
correlation functions 
$r_{j,j'}(p)=EX_j(0)X_{j'}(p)$, $1\le j,j'\le d$,
$p\in{\mathbb Z}^\nu$. In Theorem~2.2 we described 
this correlation function as the Fourier transform of a 
matrix valued spectral measure $G=(G_{j,j'})$, $1\le j,j'\le d$. 
In the case of scalar valued stationary fields there is a 
continuation of this result. A so-called random spectral 
measure $Z_G$ can be constructed, and the elements of the 
stationary random field can be represented as an appropriate 
random integral with respect to it. This result can be 
interpreted so that the elements of a scalar valued 
stationary random field can be represented as the 
Fourier transforms of a random spectral measure. We want 
to find the multi-dimensional version of this result.

The results about scalar valued stationary fields also 
help in the study of vector valued stationary random 
fields. Indeed, since the $j$-th coordinates $X_j(p)$, 
of the random vectors $X(p)$, $p\in{\mathbb Z}^\nu$, 
define a scalar valued stationary random field we can 
apply for them the results known in the scalar valued 
case. This enables us to construct such a random spectral 
measure $Z_{G.j}$ for all $1\le j\le d$ for which the 
identity $X_j(p)=\int_{[-\pi,\pi)^\nu} e^{i(p,x)}Z_{G,j}(\,dx)$ 
holds for all $p\in{\mathbb Z}^\nu$. The distribution 
of the random spectral measure $Z_{G,j}$ depends on the
coordinate $G_{j,j}$ of the matrix valued spectral 
measure $G$, which is the spectral measure of the
stationary random field $X_j(p)$, $p\in{\mathbb Z}^\nu$.  
For a fixed number $1\le j\le d$ the  properties of 
the random spectral measure $Z_{G,j}$ and the definition 
of the random integral with respect to it is worked out 
in the literature. I shall refer to my Lecture Note~\cite{11}, 
where I described this theory. 

Nevertheless, the results obtained in such a way are not 
sufficient for us. They describe the distribution of the  
random spectral measure $Z_{G,j}$ for each $1\le j\le d$, 
but we need some additional results about their joint 
behaviour. To get them I recall the results in~\cite{11} which 
led to the construction of the random spectral measures 
$Z_{G,j}$, and then I extend them in order to get the 
results we need to describe their joint distribution. 

I explain how we define simultaneously all random 
spectral measures $Z_{G,j}$, $1\le j\le d$, by recalling 
the method of~\cite{11} with some necessary modifications in 
the notation to adapt this method to our case. 

We construct the random spectral measure $Z_{G,j}$ for 
all $1\le j\le d$ in the following way. First we introduce 
two Hilbert spaces ${\cal K}^c_{1,j}$ and ${\cal H}^c_{1,j}$, 
and define an appropriate norm-preserving invertible 
linear transformation $T_j$ from ${\cal K}^c_{1,j}$ to 
${\cal H}^c_{1,j}$. (Here, and in the next discussion I 
apply the superscript $^c$ in the notation to emphasize 
that we are working in a complex, and not in a real 
Hilbert space.) The Hilbert space ${\cal K}^c_{1,j}$ 
consists of those complex  valued functions 
$u(x)$ on the torus $[-\pi,\pi)^\nu$ for which 
$\int_{[-\pi,\pi)^\nu}|u(x)|^2G_{j,j}(\,dx)<\infty$, and the 
norm is defined in this space by the formula 
$\|u\|_{0,j}^2=\int_{[-\pi,\pi)^\nu} |u(x)|^2G_{j,j}(\,dx)$.
The Hilbert space ${\cal H}^c_{1,j}$ is defined as the closure
of the linear space consisting of the linear combinations
$\sum c_{p_s}X_j(p_s)$ with some (complex valued) coefficients
$c_{p_s}$ and parameters $p_s\in{\mathbb Z}^\nu$ in the
Hilbert space ${\cal H}^c$. The Hilbert space ${\cal H}^c$
consists of the complex valued random variables 
with finite second moment, measurable with respect to 
the $\sigma$-algebra generated by the random variables 
$X_j(p)$, $1\le j\le d$, $p\in{\mathbb Z}^\nu$, and 
the norm $\|\cdot\|_{1,j}$ in it is determined by 
the scalar product defined by the formula 
$\langle\xi,\eta\rangle=E\xi\bar\eta$, 
$\xi,\eta\in{\cal H}^c$. First we define the
transformation $T_j$ only for finite trigonometrical 
sums in ${\cal K}^c_{1,j}$. We define it by the formula 
$T_j(\sum c_{p_s}e^{i(p_s,x)})=\sum c_{p_s}X_j(p_s)$. We showed 
in~\cite{11} that we have defined in such a way a norm-preserving
linear transformation from an everywhere dense subspace
of ${\cal K}^c_{1,j}$ to an everywhere dense subspace of 
${\cal H}^c_{1,j}$. This can be extended to a norm-preserving 
invertible linear transformation $T_j$ from ${\cal K}^c_{1,j}$ to 
${\cal H}^c_{1,j}$ in a unique way. We define the random spectral 
measure $Z_{G,j}(A)$ for a measurable set $A\subset[-\pi,\pi)^\nu$ 
by the formula 
$Z_{G,j}(A)=T_j({\mathbb I}_A(\cdot))$, where ${\mathbb I}_A(\cdot)$ 
denotes the indicator function of the set~$A$.

It follows from the results of~\cite{11} that for any 
$1\le j\le d$ the measure $G_{j,j}$ determines the 
distribution of the random spectral measure $Z_{G,j}$, 
(i.e. the joint distribution of the random variables 
$Z_{G,j}(A_1),\dots Z_{G,j}(A_N)$ for all $N\ge1$ and
measurable sets $A_k\subset [-\pi,\pi)^\nu$, 
$1\le k\le N$). Next we shall study the joint 
distribution of the random fields $Z_{G,j}$ for all
$1\le j\le d$, i.e. the joint distribution of the
random variables $Z_{G,j}(A_1),\dots Z_{G,j}(A_N)$ for all 
$N\ge1$, measurable sets $A_k\subset [-\pi,\pi)^\nu$, 
$1\le k\le N$ and $1\le j\le d$. In particular, we shall
show that the joint distribution of the random fields
$Z_{G,j}$, $1\le j\le d$, are determined by the matrix 
valued spectral measure $G=(G_{j,j'})$, $1\le j,j'\le d$.
The joint distribution of these random fields are 
determined by the matrix valued measure $G$, and not 
only by their diagonal elements $G_{j,j}$, $1\le j\le d$.

To investigate the joint behaviour of the random spectral
measures $Z_{G,j}$, $1\le j\le d$, first we define two
Hilbert spaces ${\cal K}^c_1$ and ${\cal H}^c_1$ together with 
a norm-preserving  and invertible transformation between 
them. The elements of the Hilbert space ${\cal K}^c_1$ are 
the vectors $u=(u_1(x),\dots,u_d(x))$ with 
$u_j(x)\in{\cal K}^c_{1,j}$, $1\le j\le d$. To define the 
(semi)-norm in ${\cal K}^c_1$ we introduce a positive 
semidefinite bilinear form $\langle \cdot,\cdot\rangle_0$ 
on it. To make some subsequent discussions simpler I 
make the following convention in the rest of the paper. 
Given a positive semidefinite matrix valued measure 
$(G_{j,j'})$, $1\le j,j'\le d$, on the torus 
$[-\pi,\pi)^\nu$, I fix a finite and even measure 
$\mu$ on $[-\pi,\pi)^\nu$ such that all complex 
measures $G_{j,j'}$ are absolutely 
continuous with respect to it, and I denote by 
$g_{j,j'}(x)$ their Radon--Nikodym derivative with 
respect to $\mu$. With the help of this notation 
we define $\langle \cdot,\cdot\rangle_0$ in the 
following way. If 
$u(x)=(u_1(x),\dots,u_d(x))\in{\cal K}^c_1$  and 
$v(x)=(v_1(x),\dots,v_d(x))\in{\cal K}^c_1$, then
\begin{eqnarray}
\langle u(x), v(x)\rangle_0
&=&\sum_{j=1}^d\sum_{j'=1}^d\int u_j(x)\overline{v_{j'}(x)}G_{j,j'}(\,dx) 
\label{3.1} \\ 
&=&\sum_{j=1}^d\sum_{j'=1}^d\int g_{j,j'}(x)u_j(x)
\overline{ v_{j'}(x)}\mu(\,dx) \nonumber \\ 
&=&\int_{[-\pi,\pi)^\nu} u(x) g(x) v(x)^*\mu(\,dx) \nonumber
\end{eqnarray}
with the matrix $g(x)=(g_{j,j'}(x))$, $1\le j,j'\le d$, where 
$v^*(x)$ denotes the column vector whose elements are
the functions $\overline {v_k(x)}$, $1\le k\le d$. 

To show that the integral in the definition of
$\langle u(x), v(x)\rangle_0$ is convergent let us observe 
that 
\begin{equation}
|g_{j,j'}(x)|^2\le g_{j,j}(x)g_{j',j'}(x) \textrm{ for almost all } x 
\textrm{ with respect to the measure }\mu \label{3.2}
\end{equation}
for all $1\le j,j'\le d$, because $g(x)$ 
is a positive semidefinite matrix for almost all~$x$. This fact 
together with the Schwarz inequality imply that
\begin{eqnarray*}
&&   \!\!\!\!\!\!\!
\left|\int_{[-\pi,\pi)^\nu} u_j(x)g_{j,j'}(x)\overline{v_{j'}(x)}
\mu(\,dx)\right| \\ 
&&\!\le\int_{[-\pi,\pi)^\nu} |u_j(x)|
\sqrt{g_{j,j}(x)g_{j',j'}(x)}|v_{j'}(x)|\mu(\,dx) \\
&&\le\! \left(\int_{[-\pi,\pi)^\nu} |u_j(x)|^2g_{j,j}(x)\mu(\,dx)\right)^{1/2}
\left(\int_{[-\pi,\pi)^\nu} |v_{j'}(x)|^2g_{j',j'}(x)\mu(\,dx)\right)^{1/2} \\
&&\!<\infty 
\end{eqnarray*}
for all pairs $1\le j,j'\le d$ and $u_j\in{\cal K}^c_{1,j}$
and $v_{j'}\in{\cal K}^c_{1,k}$. This implies that the integral
in~(\ref{3.1}) is finite. Moreover, the last inequality implies that
\begin{eqnarray}
\langle u(x),u(x)\rangle_0&\le&
\left(\sum_{j=1}^d
\left(\int_{[-\pi,\pi)^\nu} |u_j(x)|^2G_{j,j}(\,dx)\right)^{1/2} \right)^2
\nonumber \\
&\le& d\sum_{j=1}^d \int_{[-\pi,\pi)^\nu}|u_j(x)|^2G_{j,j}(\,dx)
=d\sum_{j=1}^d \|u_j\|^2_{0,j} \label{3.3}
\end{eqnarray}
for all $u(x)=(u_1(x),\dots,u_d(x))\in{\cal K}^c_1$. 

Observe that $\langle u(x),u(x)\rangle_0\ge 0$, because $g(x)$ 
is a positive semidefinite matrix, which implies that 
$u(x) g(x)u^*(x)\ge0$ for almost all $x$ with respect to 
the measure $\mu$. In such a way we can define
the norm $\|\cdot\|_0$ in ${\cal K}^c_1$ by the formula
$\|u\|_0=\langle u(x),u(x)\rangle_0$. We identify two elements
$u$ and $v$ in ${\cal K}^c_1$ if $\|u-v\|_0=0$.

Next we define the Hilbert space ${\cal H}^c_1$ with the norm
$\|\cdot\|_1$ on it. The elements of ${\cal H}^c_1$ are the
vectors $\xi=(\xi_1,\dots,\xi_d)$, where $\xi_j\in{\cal H}^c_{1,j}$,
$1\le j\le d$, and we define the norm on it by the formula
$\|\xi\|_1^2=E\left|\sum_{j=1}^d\xi_j\right|^2$ if 
$\xi=(\xi_1,\dots,\xi_d)\in{\cal H}^c_1$. It is the norm
induced by the scalar product 
$\langle\xi,\eta\rangle_1=E\left(\sum_{j=1}^d\xi_j\right)
\overline{\left(\sum_{j=1}^d\eta_j\right)}$ for
$\xi=(\xi_1,\dots,\xi_d)\in{\cal H}^c_1$ and
$\eta=(\eta_1,\dots,\eta_d)\in{\cal H}^c_1$. We identify
two elements $\xi\in{\cal H}^c_1$ and $\eta\in{\cal H}^c_1$
if $\|\xi-\eta\|_1=0$.

Observe that
\begin{eqnarray}
\|\xi\|_1^2 \! &=&   \! E\left(\sum_{j=1}^d\xi_j\right)
\overline{\left(\sum_{j'=1}^d\xi_{j'}\right)}
\le\sum_{j=1}^d\sum_{j'=1}^d (E|\xi_j|^2)^{1/2}(E|\xi_{j'}|^2)^{1/2}  
\label{3.4} \\
\! &=&\!   \left(\sum_{j=1}^d (E(|\xi_j|^2)^{1/2}\right)
\left(\sum_{j'=1}^d (E(|\xi|_{j'}^2)^2)^{1/2}\right)
\le d\sum_{j=1}^d E|\xi|_j^2=d\sum_{j=1}^k\|\xi_j\|^2_{1,j} \nonumber
\end{eqnarray}
for a vector $\xi=(\xi_1,\dots,\xi_d)\in{\cal H}^c_1$ 

We define the operator $T$ mapping from ${\cal K}^c_1$ to ${\cal H}^c_1$
by the formula 
$$
Tu=T(u_1,\dots,u_d)=(T_1u_1,\dots,T_du_d)
$$  
for 
$u=(u_1,\dots,u_d)$, $u_j\in{\cal K}^c_{1,j}$, with the help of 
the already defined operators $T_j$, $1\le j\le d$. We show 
that $Tu=T(u_1,\dots,u_d)=(T_1u_1,\dots,T_du_d)$ for 
$u=(u_1,\dots,u_d)\in{\cal K}^c_1$ is a norm preserving and 
invertible transformation from ${\cal K}^c_1$ to ${\cal H}^c_1$. To
prove this let us first observe that because of inequality~(\ref{3.3})
and Weierstrass' second approximation theorem the finite linear
combinations
$$
\left(\sum_{p\in A_N}c_{1,p}e^{i(p,x)},\dots,
\sum_{p\in A_N} c_{d,p}e^{i(p,x)}\right),
$$
where $A_N=\{p=(p_1,\dots,p_\nu)\colon\; -N\le p_s\le N, 
\textrm{ for all }1\le s\le \nu\}$,
constitute an everywhere dense linear subspace in ${\cal K}^c_1$,
and because of the inequality~(\ref{3.4}) the finite linear combinations
\begin{eqnarray}
&&\left(\sum_{p\in A_N} c_{1,p}X_1(p),\dots,\sum_{p\in A_N} c_{d,p}X_d(p)\right)
\nonumber \\
&&\qquad =T\left(\sum_{p\in A_N}c_{1,p}e^{i(p,x)},\dots,
\sum_{p\in A_N}c_{d,p}e^{i(p,x)}\right)
\label{3.5}
\end{eqnarray}
constitute an everywhere dense linear subspace in ${\cal H}^c_1$
if $N=1,2,\dots$, and the coefficients $c_{j,p}$, $1\le j\le d$,
$p\in A_N$, are arbitrary complex numbers. Hence the following
calculation implies that $T$ is a norm preserving and invertible
transformation from ${\cal K}^c_1$ to ${\cal H}^c_1$. 

If
$$
u(x)=\left(\sum_{p\in A_N}c_{1,p}e^{i(p,x)},\dots,
\sum_{p\in A_N}c_{d,p}e^{i(p,x)}\right)
$$
and 
$$
v(x)=\left(\sum_{p\in A_N}c'_{1,p}e^{i(p,x)},\dots,
\sum_{p\in A_N}^Nc'_{d,p}e^{i(p,x)}\right),
$$
then
\begin{eqnarray*}
&&\langle u(x),v(x)\rangle_0
=\left\langle\left(\sum_{p\in A_N} c_{1,p}e^{i(p,x)},\dots,
\sum_{p\in A_N}c_{d,p}e^{i(p,x)}\right),\right.  \\
&&\qquad \qquad \qquad\qquad \left.\left(\sum_{p\in A_N}c'_{1,p}e^{-i(p,x)},\dots,
\sum_{p\in A_N} c'_{d,p}e^{-i(p,x)}\right)
\right\rangle_0 \\
&&\qquad =\sum_{j=1}^d\sum_{j'=1}^d \sum_{s\in A_N}\sum_{t\in A_N} 
c_{j,s}\bar {c'}_{j',t}
\int_{[-\pi,\pi)^\pi} g_{j,j'}(x) e^{i(s-t,x)}\mu(\,dx) \\
&&\qquad=E\left(\sum_{j=1}^d\sum_{s\in A_N} c_{j,s}X_j(s)\right)
\overline{\left(\sum_{j'=1}^d\sum_{t\in A_N} c'_{j',t}X_{j'}(t)\right)}
=\langle Tu(x), Tv(x)\rangle_1.
\end{eqnarray*}

We shall define the random variables $Z_{G,j}(A)$ for all
indices $1\le j\le d$ and measurable sets 
$A\subset [-\pi,\pi)^\nu$, by the formula 
$Z_{G,j}(A)=T_j({\mathbb I}_A(x))$ with the above defined 
operators $T_j$, $1\le j\le d$, where ${\mathbb I}_A(\cdot)$ 
denotes the indicator function of the set 
$A\subset[-\pi,\pi)^\nu$. Next I formulate some properties 
of this class of random variables. These properties will 
appear in the definition of random spectral measures. All 
sets appearing in the next statements are measurable 
subsets of the torus $[-\pi,\pi)^\nu$. 

\medskip
\begin{description}
\item[{\rm(i)}] The random variables  $Z_{G,j}(A)$ are complex 
valued, and their real and imaginary parts are jointly Gaussian, 
i.e. for any positive integer $N$ and sets $A_s$, $1\le s\le N$, 
the random variables $\textrm{Re}\, Z_{G,j}(A_s)$, 
$\textrm{Im}\, Z_{G,j}(A_s)$, $1\le s\le N$, 
$1\le j\le d$, are jointly Gaussian.
\item[{\rm(ii)}] $EZ_{G,j}(A)=0$ for all $1\le j\le d$ and $A$,
\item[{\rm(iii)}] $EZ_{G,j}(A)\overline {Z_{G,{j'}}(B)}=G_{j,j'}(A\cap B)$
for all $1\le j,j'\le d$ and sets $A,B$.
\item[{\rm(iv)}] $\sum\limits_{s=1}^nZ_{G,j}(A_s)
=Z_{G,j}\left(\bigcup\limits_{s=1}^n A_s\right)$ if
$A_1,\dots,A_n$ are disjoint sets, $1\le j\le d$.
\item[{\rm(v)}] $Z_{G,j}(A)=\overline{Z_{G,j}(-A)}$ for all 
$1\le j\le d$ and sets $A$.
\end{description}

\medskip
Properties (i)--(v) were proved in the one-dimensional case 
e.g. in~\cite{11}. The only difference in checking its several 
dimensional version is that we have to apply the 
multi-dimensional operator $T$ from ${\cal K}^c_1$ to 
${\cal H}^c_1$ to prove property~(i), and to apply the same 
mapping~$T$ in proving~Property ~(iii). Here we exploit 
that $\langle u,v\rangle_0=\langle Tu,Tv\rangle_1$. We apply 
this identity with the vector $u\in{\cal K}^c_1$ whose 
$j$-th coordinate is ${\mathbb I}_A(x)$, and the other coordinates 
are zero and  the vector $v\in{\cal K}^c_1$ whose $k$-th 
coordinate is ${\mathbb I}_B(x)$ and the other coordinates are zero. 
Property~(v) can be proved as the special case of the 
following more general relation.

\medskip
\begin{description}
\item[{\rm(v$'$)}] $T_j(u)=\overline{T_j(u_-)}$ for all $1\le j\le d$ 
and $u\in{\cal K}^c_j$, where $u_-(x)=\overline{u(-x)}$.
\end{description}
\medskip

Property~(v$'$) can be proved by first proving it in the special
case when $u(x)$ is a trigonometrical polynomial, and then 
applying a limiting procedure.

Next we define the vector valued random spectral measures 
corresponding to a matrix valued spectral measure. 

\medskip\noindent
{\bf Definition of vector valued random spectral measures on 
the torus.} {\it Let a matrix valued spectral measure 
$G=(G_{j,j'})$, $1\le j,j'\le d$, be given on the torus 
$[-\pi,\pi)^\nu$ together with a set of complex valued 
random variables indexed by pairs $(j,A)$, where $1\le j\le d$, 
and $A$ is an element of the $\sigma$-algebra~${\cal A}$ 
$$
{\cal A}=\{ A\colon\; A\subset [-\pi,\pi)^\nu 
\textrm{ is a Borel measurable set}\}
$$ 
of the Borel measurable sets of the torus 
whose joint distribution depends on the matrix valued spectral 
measure~$G$. To recall this dependence we denote the random
variable indexed by a pair $(j,A)$, $1\le j\le d$, $A\in {\cal A}$,
by $Z_{G,j}(A)$. We call the set of random variables $Z_{G,j}(A)$,
$1\le j\le d$, $A\in{\cal A}$, a $d$-dimensional vector valued 
random spectral measure corresponding to the matrix valued 
spectral measure~$G$ on the torus $[-\pi,\pi)^\nu$ if this set 
of random variables satisfies properties~(i)--(v) defined above. 
Given a fixed parameter $1\le j\le d$ we call the set of random 
variables $Z_{G,j}(A)$, $A\in{\cal A}$, the $j$-th coordinate of 
this $d$-dimensional vector valued random spectral measure, and 
we denote it by $Z_{G,j}$. We denote the vector valued random 
spectral measure $Z_{G,j}(A)$, $1\le j\le d$, $A\in{\cal A}$, 
by $Z_G=(Z_{G,1},\dots,Z_{G,d})$. 

More generally, if a matrix valued spectral measure $G$ is given 
on the torus $[-B,B)^\nu$ with some number $B>0$ together with a
set of complex valued random variables $Z_{G,j}(A)$, where
$1\le j\le d$, and $A$ is a Borel measurable set on the torus
$[-B,B)^\nu$ which satisfies properties (i)--(v) defined above,
then we call this set of random variables a $d$-dimensional vector
valued random spectral measure corresponding to the spectral 
measure~$G$. We call the set of random variables 
$Z_{G,j}(A)$, $A\in{\cal A}$, for a fixed $1\le j\le d$ the $j$-th
coordinate of this vector valued spectral measure, and denote it
by~$Z_{G,j}$. We denote the vector valued spectral measure by
$Z_G=(Z_{G,1},\dots,Z_{G,d})$.}

\medskip\noindent
{\it Remark:}\/ If $G=(G_{j,j'})$, $1\le j,j'\le d$, is a 
matrix valued spectral measure, $Z_G=(Z_{G,1},\dots,Z_{G,d})$ 
is a vector valued spectral measure corresponding to it, 
then $G_{j,j}$ is a scalar valued spectral measure for any 
$1\le j\le d$, and $Z_{G,j}$ is a scalar valued random 
spectral measure corresponding to it.

\medskip
It follows from the above considerations that for any 
$d$-dimensional matrix valued spectral measure there exists 
a $d$-dimensional vector valued random spectral measure 
corresponding to it. We can define the random integral with 
respect to it by means of the method applied in the scalar 
valued case.

We shall define the random integrals of the functions
$f\in{\cal K}^c_{1,j}$ with respect to the random spectral
measure $Z_{G,j}$, $1\le j\le d$. First  we define these
integrals for elementary functions. They are finite sums of 
the form $\sum_{s=1}^N c_s {\mathbb I}_{A_s}(x)$, where 
$A_1,\dots,A_N$ are disjoint sets in $[-\pi,\pi)^\nu$, and 
$c_s$, $1\le s\le N$, are arbitrary complex numbers. Their 
integrals with respect to the random spectral measure 
$Z_{G,j}$, $1\le j\le d$, are defined as
$$
\int\left(\sum_{s=1}^N c_s {\mathbb I}_{A_s}(x)\right)Z_{G,j}(\,dx)
=\sum_{s=1}^N c_s Z_{G,j}(A_s).
$$
As it is remarked in~\cite{11}, property~(iv) implies that this definition
is meaningful, the integral of an elementary function does not depend
on its representation.  Then a simple calculation with the help of~(iii) 
shows that for two elementary functions $u$ and $v$
\begin{equation}
E\left(\int u(x)Z_{G,j}(\,dx)\overline{\int v(x)Z_{G,j}(\,dx)}\right)
=\int u(x)\overline{v(x)}G_{j,j}(\,dx), \quad  1\le j\le d. \label{3.6}
\end{equation}
This implies that the integral of the elementary functions with 
respect to the random spectral measure $Z_{G,j}$ define a norm 
preserving transformation from an everywhere dense subspace of 
the Hilbert space of ${\cal K}^c_{1,j}$ to an everywhere dense 
subspace of the Hilbert space of ${\cal H}^c_{1,j}$. This can be 
extended to a unitary transformation from ${\cal K}^c_{1,j}$ to 
${\cal H}^c_{1,j}$ in a unique way, and this extension defines 
the integral of a function $u\in{\cal K}^c_{1,j}$. It is clear 
that relation~(\ref{3.6}) remains valid for general functions 
$u,v\in{\cal K}^c_{1,j}$. Moreover, it is not difficult to see 
with the help of~(iii) that it can be generalized to the formula
\begin{equation}
E\left(\int u(x)Z_{G,j}(\,dx)\overline{\int v(x)Z_{G,{j'}}(\,dx)}\right)
=\int u(x)\overline{v(x)}G_{j,j'}(\,dx) \label{3.7}
\end{equation}
if $u\in{\cal K}^c_{1,j}$ and $v\in{\cal K}^c_{1,j'}$, $1\le j,j'\le d$.

It is clear that
\begin{equation}
E\int u(x) Z_{G,j}(\,dx)=0 \quad\textrm{for all } u\in{\cal K}_{1,j},
\quad 1\le j\le d.  \label{3.8}
\end{equation}

Another important property of the random integrals with respect to
$Z_{G,j}$ is that for all $1\le j\le d$
\begin{equation}
\int u(x)Z_{G,j}(\,dx) \quad\textrm {is real valued if } 
u(-x)=\overline{u(x)} \textrm{ for $\mu$ almost all }x\in[-\pi,\pi)^\nu. 
\label{3.9}
\end{equation}
This relation holds, since
$\int u(x)Z_{G,j}(\,dx) =\overline{\int u(x)Z_{G,j}(\,dx)}$
if $u(-x)=\overline{u(x)}$. We get this identity by means of the
change of variables $x\to -x$ with the help of relation~(v).

In the next Theorem I formulate the results we have about random
spectral measures and random integrals with respect to them.

\medskip\noindent
{\bf Theorem 3.1} {\it Given a positive semidefinite matrix 
valued, even measure $G=(G_{j,j'})$, $1\le j,j'\le d$, on 
the torus $[-\pi,\pi)^\nu$ there exists a vector valued 
random spectral measure $Z_G=(Z_{G,1},\dots,Z_{G,d})$ 
corresponding to it. We have defined the random integrals 
$\int u(x)Z_{G,j}(\,dx)$ for all $1\le j\le d$ and 
$u\in{\cal K}^c_{1,j}$. This is a linear operator which 
satisfies relations~(\ref{3.7}), (\ref{3.8}), (\ref{3.9}), 
and the formula
\begin{equation}
X_j(p)=\int_{[-\pi,\pi)^\nu} e^{i(p,x)}Z_{G,j}(\,dx),  
\quad 1\le j\le d, \;\; p\in{\mathbb Z}^\nu, \label{3.10}
\end{equation}
defines a $d$-dimensional vector valued Gaussian stationary 
field whose matrix valued spectral measure is $G=(G_{j,j'})$, 
$1\le j,j'\le d$. Moreover, if a $d$-dimensional vector 
valued Gaussian stationary random field is given with this 
matrix valued spectral measure, then the random integrals 
in formula~(\ref{3.10}) taken with respect to the random spectral 
measure that we have constructed with its help through 
an operator $T$ in this section equals this vector valued 
Gaussian stationary random field.}

\medskip\noindent
{\it Proof of Theorem 3.1.} We have already proved the 
existence  of the vector valued random spectral measure, 
and we constructed the random integral with respect to it. 
It satisfies formulas~(\ref{3.7}) and~(\ref{3.8}).  The random 
variables $X_j(p)$ defined in~(\ref{3.10}) are real valued 
by~(\ref{3.9}) and Gaussian with expectation zero. Hence we can 
show that they define a Gaussian stationary sequence by 
calculating their correlation function. We get by formula~(\ref{3.7}) 
that $EX_j(p)X_{j'}(q)=\int_{[-\pi,\pi)^\nu} e^{i(p-q,x)}G_{j,j'}(\,dx)$,
and this had to be checked. If the random spectral measure is 
constructed in the way as we have done in this section, then a 
comparison of the random integral we have defined with its help 
and of the operator~$T$ shows that 
$\int u(x)Z_{G,j}(\,dx)=T_j(u(x))$ for all $u\in{\cal K}^c_{1,j}$.
In particular,
$\int_{-[\pi,\pi)^\nu} e^{i(p,x)}Z_{G,j}(\,dx)=T_j(e^{i(p,x)})=X_j(p)$. 
This identity implies the last relation of Theorem~3.1. 
Theorem~3.1 is proved.

\medskip
Formula (\ref{3.9}) and Theorem~3.1 make possible to define for all
$1\le j\le d$ a real Hilbert space ${\cal K}_{1,j}$ consisting
of appropriate elements of ${\cal K}^c_{1,j}$ for which the operator
$T_j$ is a norm preserving invertible transformation from
${\cal K}_{1,}j$ to the real Hilbert space ${\cal H}_{1,j}$ consisting
of the real valued functions of the Hilbert space ${\cal H}^c_{1,j}$.
More precisely, the following statement holds.

\medskip\noindent
{\bf Lemma 3.2.} {\it Let $(G_{j,j'})$, $1\le j,j'\le d$, be a
matrix valued spectral measure on the torus $[-\pi,\pi)^\nu$,
and let $(Z_{G,1},\dots,Z_{G,d})$ be a vector valued spectral
measure corresponding to it. Define the $d$-dimensional vector
valued Gaussian stationary field $(X_1(p),\dots,X_p(d))$ by
formula~(\ref{3.10}) with the help of this vector valued random 
spectral measure. Define for all $1\le j\le d$ the  set of 
complex valued functions ${\cal K}_{1,j}$ on the torus 
$[-\pi,\pi)^\nu$ as
$$
{\cal K}_{1,j}=\left\{u\colon \int |u(x)|^2G_{j,j}(\,dx)<\infty, \quad 
u(-x)=\overline{u(x)}\textrm{ for all } x\in[-\pi,\pi)^\nu\right\}.
$$
Then ${\cal K}_{1,j}$ is a real Hilbert space with the scalar product
$$
\langle u,v\rangle=\int u(x)\overline{v(x)}G_{j,j}(\,dx), 
\quad u,v\in{\cal K}_{1,j}.
$$ 
Let ${\cal H}_{1,j}$ be the real Hilbert 
space consisting of the closure of the finite linear combinations 
$\sum_{k=1}^Nc_kX_j(p_k)$, $p_k\in{\mathbb Z}^\nu$, with real
coefficients~$c_k$ in the Hilbert space ${\cal H}$ of  random 
variables with finite second moments in the probability space 
where the random spectral measures $Z_{G,j}$ exists. (We define 
the scalar product in ${\cal H}$ in the usual way.) Then the 
map $T_j(u)=\int u(x)Z_{G,j}(\,dx)$, $u\in{\cal K}_{1,j}$, is a 
norm preserving, invertible linear transformation from 
the real Hilbert space ${\cal K}_{1,j}$ to the real 
Hilbert space~${\cal H}_{1,j}$.}

\medskip\noindent
{\it Proof of Lemma 3.2.}\/ The space ${\cal K}_{1,j}$ is a 
real Hilbert space, since the change of variable $x\to -x$ 
in the integral
$\langle u,v\rangle=\int u(x)\overline{v(x)}G_{j,j}(\,dx)$ 
implies that
$\langle u,v\rangle=\overline{\langle u,v\rangle}$ for all 
$u,v\in{\cal K}_{1,j}$ because of the evenness of the measure 
$G_{j,j}$. Clearly $e^{i(p,x)}\in{\cal K}_{1,j}$ for all 
$p\in{\mathbb Z}^\nu$. The class of functions 
${\cal K}_{1,j}$ agrees with the class of functions which 
have the form $u(x)=\frac{v(x)+\overline{v(-x)}}2$
with some $v\in{\cal K}^c_{1,j}$. As a consequence the set 
of finite trigonometrical polynomials $\sum c_ke^{i(p_k,x)}$, 
$p_k\in{\mathbb Z}^\nu$, with real valued coefficients $c_k$ 
is an everywhere dense subspace of ${\cal K}_{1,j}$. Since 
$T_j(\sum c_ke^{i(p_k,x)})=\sum c_k X_j(p_k)$, the transformation 
$T_j$ maps an everywhere dense subspace of ${\cal K}_{1,j}$ to
an everywhere dense subspace of ${\cal H}_{1,j}$. Because of
formulas (\ref{3.7}) and~(\ref{3.9}) $T_j$ is a norm preserving 
transformation in ${\cal K}_{1,j}$. Hence $T_j$ is an invertible, 
norm preserving transformation from ${\cal K}_{1,j}$ to 
${\cal H}_{1,j}$. Lemma~3.2 is proved.

\medskip
I would remark that the transformation $T_j$ on ${\cal K}_{1,j}$ 
defined in Lemma~3.2 is the restriction of the previously 
defined transformation $T_j$ on ${\cal K}^c_{1,j}$ to its subset 
${\cal K}_{1,j}$. I make also the following remark.

\medskip\noindent
{\bf Lemma 3.3.} {\it The positive semidefinite matrix valued,
even  measure $G(A)=(G_{j,j'}(A))$, $1\le j,j'\le d$, 
$A\in[-\pi,\pi)^\nu$, determines the distribution of a 
vector valued spectral random measure $Z_{G,j}$, $1\le j\le d$, 
corresponding to it.}

\medskip\noindent
To prove this lemma we have to show that for any collection of 
measurable sets $A_1$,\dots, $A_N$, the matrix valued measure 
$G(A)$ determines the joint distribution of the random vector 
consisting of the elements 
$\textrm{Re}\,Z_{G,j}(A_s)$, $\textrm{Im}\,Z_{G,j}(A_s)$, $1\le s\le N$,
$1\le j\le d$. Since this is a Gaussian random vector with 
expectation zero, it is enough to check that the covariance 
of these random variables can be expressed by means of the 
matrix valued measure $G(A)$. Since 
$\textrm{Re}Z_{G,j}(A)=\frac{Z_{G,j}(A)+\overline{Z_{G,j}(A)} }2$ 
and $\textrm{Im }Z_{G,j}(A)=\frac{Z_{G,j}(A)-\overline{Z_{G,j}(A)} }{2i}$ 
we can calculate these covariances with the help of 
properties~(iii) and~(v) of vector valued random spectral measures.

\medskip
Finally I prove an additional property of the vector valued 
random spectral measures which will be useful in the study of 
multiple Wiener--It\^o integrals.

\medskip
\begin{description}
\item[{\rm(vi)}] The random variables of the form $Z_{G,j}(A\cup(-A))$ 
are real valued. Let a set $A\cup(-A)$ be disjoint from some sets
$B_1\cup(-B_1)$,\dots, $B_n\cup(-B_n)$. Then for any indices 
$1\le j,j'\le d$ the (complex valued) random vector 
$(Z_{G,j}(A), Z_{G,{j'}}(A))$, is independent of the random vector 
consisting of the elements $Z_{G,k}(B_s)$, $1\le s\le n$, $1\le k\le d$.
\end{description}

\medskip\noindent
{\it Proof of property (vi).}\/ It follows from property~(v) 
that $Z_{G,j}(A\cup(-A))=\overline{Z_{G,j}(A\cup(-A))}$, hence 
$Z_{G,j}(A\cup(-A))$ is real valued. To prove the second 
statement of~(vi) it is enough to check that under its conditions 
the (real valued) random variables $\textrm{Re}\,Z_{G,j}(A)$ 
and $\textrm{Im}\,Z_{G,j}(A)$ are uncorrelated to all random 
variables $\textrm{Re}\,Z_{G,k}(B_s)$, $\textrm{Im}\,Z_{G,k}(B_s)$,
$1\le s\le n$, $1\le k\le d$. This relation holds, since 
by the conditions of~(vi) $(\pm A)\cap(\pm B_s)=\emptyset$, 
hence relation~(iii) implies that
$EZ_{G,j}(\pm A)\overline{Z_{G,{j'}}(\pm B_s)}=0$ for all sets $B_s$, 
$1\le s\le n$, and indices $1\le j,j'\le d$. On the other hand, 
all covariances can be expressed as a linear combination of 
such expressions, since by relation~(v) $\textrm{Re}\,Z_{G,j}(\pm A)= 
\frac{Z_{G,j}(\pm A)+\overline{Z_{G,j}(\pm A)}}2
=\frac{Z_{G,j}(\pm A)+ Z_{G,j}(\mp A)}2$, and a similar relation 
holds also for $\textrm{Im}\,Z_{G,j}(\pm A)$,
$\textrm{Re}\,Z_{G,{j'}}(\pm B_s)$ and $\textrm{Im}\,Z_{G,{j'}}(\pm B_s)$,
$1\le s\le n$, $1\le j'\le d$. 

\section{Spectral representation of vector valued stationary 
generalized random fields}

In Sections 2 and 3 we discussed the properties of vector valued 
Gaussian stationary random fields with discrete parameters, which 
means a class of Gaussian random vectors $X(p)$, $p\in{\mathbb Z}^\nu$, 
with some nice properties. Similarly, we could have defined and 
investigated vector valued Gaussian stationary random fields 
with continuous parameters, where we consider a set of random 
vectors $X(t)$ indexed by  $t\in {\mathbb R}^\nu$ which have some nice 
properties. But we do not discuss this topic here. Here we 
define and investigate instead so-called vector valued 
Gaussian stationary generalized random fields  
$X(\varphi)=(X_1(\varphi),\dots,X_d(\varphi))$, parametrized 
with a nice linear space of functions~$\varphi$. 

Actually I am interested here in the vector valued
Gaussian stationary generalized random fields not for 
their own sake. We shall construct a class of vector 
valued, Gaussian stationary generalized random fields. 
We shall show that their distribution can be described 
by means of a matrix valued spectral measure. We can 
also construct a vector valued random spectral measure 
in such a way that the elements of our vector valued 
generalized random field can be expressed in a form that 
can be considered as the Fourier transform of 
this random spectral measure. These matrix valued 
spectral measures and vector valued random spectral 
measures slightly differ from those defined in 
Sections~2 and~3, but since they are very similar to 
the corresponding objects defined for stationary 
random fields with discrete parameters it is natural 
to give them the same name.

The results that we shall prove are very similar to 
the results we got about vector valued random fields with 
discrete parameters. The main difference is that we can 
construct a larger class of matrix valued spectral 
measures and vector valued random spectral measures by 
means of generalized random fields. We shall need them, 
because in our later investigations we shall deal with 
such limit theorems where we can express the limit by 
means of these new, more general objects. On the other 
hand, these new vector valued random spectral measures 
behave similarly to the previous ones. In particular, 
the later results of this paper about multiple
Wiener--It\^o integrals also hold for this more general 
class of vector valued random spectral measures. Let 
me remark that we met a similar picture in the study 
of scalar valued Gaussian random fields in~\cite{11}, 
so that here we actually generalize the results in 
that work to the multi-dimensional case.

In the definition of vector valued generalized random 
fields we shall choose the functions of the Schwartz 
space for the class of parameter set. So to define the 
vector valued generalized random fields first I recall 
the definition of the Schwartz space, (see~\cite{7}).

We define the Schwartz space ${\cal S}$ of real valued 
functions on ${\mathbb R}^\nu$ together with its version 
${\cal S}^c$ consisting of complex valued functions on 
${\mathbb R}^\nu$. The space ${\cal S}^c=({\cal S}^\nu)^c$  
consists of those complex valued functions of $\nu$ 
variables which decrease at infinity, together with 
their derivatives, faster than any polynomial degree. 
More explicitly, $\varphi\in{\cal S}^c$ for a complex 
valued function $\varphi$ defined on ${\mathbb R}^\nu$ if
$$
\left|x_1^{k_1}\cdots x_\nu^{k_\nu}\frac{\partial^{q_1+\cdots+q_\nu}}
{\partial x_1^{q_1}\dots \partial x_\nu^{q_\nu}}
\varphi(x_1,\dots,x_\nu)\right|
\le C(k_1,\dots,k_\nu,q_1,\dots,q_\nu)
$$
for all points $x=(x_1,\dots,x_\nu)\in {\mathbb R}^\nu$ and  vectors
$(k_1,\dots,k_\nu)$, $(q_1,\dots,q_\nu)$ with non-negative
integer coordinates with some constant
$C(k_1,\dots,k_\nu,q_1,\dots,q_\nu)$ which may depend on the
function~$\varphi$. The elements of the space ${\cal S}$ 
are defined similarly, with the only difference that they 
are real valued functions.

To complete the definition of the spaces ${\cal S}$ 
and ${\cal S}^c$ we still have to define the topology
in them. We introduce the following topology in these
spaces. 

Let a basis of neighbourhoods of the origin consist of 
the sets
$$
U(k,p,\varepsilon)=\left\{\varphi\colon\; \varphi\in{\cal S}, \;\;
\max_{\substack{q=(q_1,\dots,q_\nu)\\
0\le q_s\le p,\textrm{ for all } 1\le s\le \nu}}
\sup_x(1+|x|^2)^k |D^q\varphi(x)|<\varepsilon\right\}
$$
with $k=0,1,2,\dots$, $p=1.2,\dots$ and $\varepsilon>0$, 
where $|x|^2=x_1^2+\cdots+x_\nu^2$, and 
$D^q=\frac{\partial^{q_1+\cdots+q_\nu}}
{\partial x_1^{q_1}\dots \partial x_\nu^{q_\nu}}$
for $q=(q_1,\dots,q_\nu)$. A basis of neighbourhoods of an 
arbitrary function $\varphi\in{\cal S}^c$ (or 
$\varphi\in{\cal S}$) consists of sets of the form 
$\varphi+U(k,q,\varepsilon)$, where the class of sets 
$U(k,q,\varepsilon)$ is a basis of neighbourhood of the 
origin. Actually we shall use only the following property 
of this topology. A sequence of functions 
$\varphi_n\in{\cal S}^c$ (or $\varphi_n\in{\cal S}$) 
converges to a function $\varphi$ in this topology if and 
only if
$$
\lim_{n\to\infty}\sup_{x\in {\mathbb R}^\nu}
(1+|x|^2)^k|D^q\varphi_n(x)-D^q\varphi(x)|=0.
$$
for all $k=1,2,\dots$ and $q=(q_1,\dots,q_\nu)$.
The limit function $\varphi$ is also in the
space~${\cal S}^c$ (or in the space ${\cal S}$).

I shall define the notion of vector valued generalized 
random fields together with some related notions with 
the help of the notion of Schwartz spaces. A 
$d$-dimensional generalized random field is a random 
field whose elements are $d$-dimensional random vectors
$$
(X_1(\varphi),\dots,X_d(\varphi))=
(X_1(\varphi,\omega),\dots,X_d(\varphi,\omega))
$$ 
defined for all functions $\varphi\in {\cal S}$,
where ${\cal S}={\cal S}^\nu$ is the Schwartz space.
Before defining vector valued generalized
random fields I write down briefly the idea of 
their definition. This is explained in~\cite{11} 
and~\cite{12} in more detail.

Given a vector valued Gaussian stationary random 
field 
$$
X(t)=(X_1(t),\dots,X_d(t)), \quad t\in {\mathbb R}^\nu,
$$ 
we can define with its help the random field 
$X(\varphi)=(X_1(\varphi),\dots,X_d(\varphi))$,
$\varphi\in{\cal S}^\nu$, 
$X_j(\varphi)=\int\varphi(t)X_j(t)\,dt$, $1\le j\le d$, 
indexed by the elements of the Schwartz space, and 
this determines the original random field. We define 
generalized random fields with elements indexed by 
$\varphi\in{\cal S}$ as such random fields which
behave similarly to the random fields defined by
means of such integrals. 


\medskip\noindent
{\bf Definition of vector valued generalized random fields.} 
{\it We say that the set of random vectors 
$(X_1(\varphi),\dots,X_d(\varphi))$,  $\varphi\in{\cal S}$, 
is a $d$-dimensional vector valued generalized random field 
over the Schwartz space ${\cal S}={\cal S}^\nu$ of rapidly
decreasing smooth functions if:

\medskip
\begin{description}
\item[{\rm(a)}]  $X_j(a_1\varphi+a_2\psi)=a_1X_j(\varphi)+
a_2X_j(\psi)$ with probability 1 for the $j$-th coordinate of 
the random vectors $(X_1(\varphi),\dots,X_d(\varphi))$ and  
$(X_1(\psi),\dots,X_d(\psi))$. This relation holds for each 
coordinate $1\le j\le d$, all real numbers $a_1$ and $a_2$,
 and pair of functions $\varphi$, $\psi$  from the Schwartz 
space ${\cal S}$. (The exceptional set of probability~0 
where this identity does not hold may depend on $a_1$, 
$a_2$, $\varphi$ and $\psi$.)

\item[{\rm(b)}] $X_j(\varphi_n)\Rightarrow X_j(\varphi)$ 
stochastically for any $1\le j\le d$ if 
$\varphi_n\to\varphi$ in the topology of ${\cal S}$.
\end{description}
}

\medskip
We also introduce the following definition. In its formulation 
we use the notation $\stackrel{\Delta}{=}$ for equality in 
distribution. 

\medskip\noindent
{\bf Definition of stationarity and Gaussian property for a vector 
valued generalized random field.} 
{\it The $d$-dimensional vector valued generalized random field
$X=\{(X_1(\varphi)\dots,X_d(\varphi)),\;\varphi\in {\cal S}\}$ 
is stationary if 
$$
(X_1(\varphi)\dots,X_d(\varphi))\stackrel{\Delta}{=}
(X_1(T_t\varphi)\dots,X_d(T_t\varphi))
$$
for all $\varphi\in{\cal S}$ and $t\in {\mathbb R}^\nu$, where 
$T_t\varphi(x)=\varphi(x-t)$. It is Gaussian
if $(X_1(\varphi),\dots,X_d(\varphi))$ is a Gaussian random 
vector for all $\varphi\in{\cal S}$. We call a vector valued
generalized random field a vector valued generalized random field
with zero expectation if $EX_j(\varphi)=0$ for all 
$\varphi\in{\cal S}$ and coordinates $1\le j\le d$.} 

\medskip
In the definition of stationarity and Gaussian property we 
imposed a condition for a single random vector. But because 
of the linearity property of generalized random fields 
formulated in property~(a) of their definition and the fact 
that if we have $N$ random vectors $\xi_1,\dots,\xi_N$ and 
$\eta_1,\dots,\eta_N$ such that the linear combinations 
$\sum\limits_{k=1}^Na_k\xi_k$ and $\sum\limits_{k=1}^Na_k\eta_k$ 
have the same distribution for any coefficients $a_k$, 
$1\le k\le N$, then the joint distribution of the random 
vectors $\xi_1,\dots,\xi_N$ and $\eta_1,\dots,\eta_N$ 
agree imply that an analogous statement holds about the 
properties of the joint distribution of several random 
vectors in a vector valued stationary random field. 
Indeed, if we take $N$ random vectors 
$(X_1(\varphi_k),\dots,X_d(\varphi_k))$, $1\le k\le N$, 
then their joint distribution agrees with the joint 
distribution of their shifts 
$(X_1(T_t\varphi_k),\dots,X_d(T_t\varphi_k))$, 
$1\le k\le N$, for any $t\in {\mathbb R}^\nu$. This follows 
from the fact that
$$
\sum_{k=1}^N a_k (X_1(\varphi_k),\dots,X_d(\varphi_k))
\stackrel{\Delta}{=} \sum_{k=1}^N a_k 
(X_1(T_t\varphi_k),\dots,X_d(T_t\varphi_k))
$$ 
for all $t\in {\mathbb R}^\nu$ and coefficients $a_k$, 
$1\le k\le N$, for a $d$-dimensional vector valued 
stationary generalized random field because of the 
linearity property of the generalized random fields 
and the properties of the operator $T_t$. A similar 
argument shows that the joint distribution of some 
vectors
$(X_1(\varphi_k),\dots,X_d(\varphi_k))$, $1\le k\le N$, 
in a
vector valued Gaussian generalized random field 
is Gaussian. 

\medskip
I shall construct a large class of $d$-dimensional vector
valued Gaussian stationary generalized random fields with 
expectation zero. I shall construct them with the help 
of positive semidefinite matrix valued  even measures on 
${\mathbb R}^\nu$. In the next step I write down this definition. 
The main difference between the definition of this notion 
and its counterpart defined on the torus $[-\pi,\pi)^\nu$ 
is that now we consider such complex  
measures which may have non-finite total variation. We 
impose instead a less restrictive condition. We shall 
work with complex measures on ${\mathbb R}^\nu$ which 
have locally finite total variation. For the sake of 
completeness I give their definition.

\medskip
\noindent
{\bf Definition of complex measures on ${\mathbb R}^\nu$
with locally finite total variation. The definition of their
evenness property.} {\it A complex measure on 
${\mathbb R}^\nu$ with locally finite total variation is such a complex 
valued function on the bounded, Borel measurable 
subsets of ${\mathbb R}^\nu$ whose restrictions to the measurable 
subsets of a cube $[-T,T]^\nu$ are complex  
measures with finite total variation for all $T>0$. We say
that a complex measure $G$ on ${\mathbb R}^\nu$ with 
locally finite total variation is even, if $G(-A)=\overline{G(A)}$ 
for all bounded and measurable sets $A\subset {\mathbb R}^\nu$.} 

\medskip
Let me remark that not all complex measures with 
locally finite total variation can be extended to a complex 
measure on all measurable subsets of ${\mathbb R}^\nu$. On the other 
hand, this can be done if we are working with a (real, 
positive number valued) measure. Next I formulate the 
definition we need in our discussion.

\medskip\noindent
{\bf Definition of positive semidefinite matrix valued 
measures on ${\mathbb R}^\nu$ with moderately increasing distribution 
at infinity. The definition of their evenness property.} 
{\it A Hermitian matrix valued measure on ${\mathbb R}^\nu$ is a 
class of such Hermitian matrices $(G_{j,j'}(A))$, 
$1\le j,j'\le d$, defined for all bounded, measurable sets 
$A\subset {\mathbb R}^{\nu}$ for which all coordinates $G_{j,j'}(\cdot)$,
$1\le j,j'\le d$, are complex measures on ${\mathbb R}^\nu$ 
with locally finite total variation. We call a Hermitian matrix 
valued measure $(G_{j,j'}(\cdot))$, $1\le j,j'\le d$, on ${\mathbb R}^\nu$ 
positive semidefinite if there exists a ($\sigma$-finite) 
positive measure $\mu$ on ${\mathbb R}^\nu$ such that for all numbers 
$T>0$ and indices  $1\le j,j'\le d$ the restriction of the 
complex measures $G_{j,j'}$ to the cube 
$[-T,T]^\nu$ is absolutely continuous with respect to $\mu$, 
and the matrices  $(g_{j,j'}(x))$, $1\le j,j'\le d$, defined 
with the help of the Radon--Nikodym  derivatives 
$g_{j,j'}(x)=\frac{dG_{j,j'}}{d\mu}(x)$, $1\le j,j'\le d$,
are Hermitian, positive semidefinite matrices for almost 
all $x\in {\mathbb R}^\nu$ with respect to the measure~$\mu$. We call
this Hermitian matrix valued measure $(G_{j,j'}(\cdot))$,
$1\le j,j'\le d$, on ${\mathbb R}^\nu$ even if the complex  
measures $G_{j,j'}$ with locally finite variation 
are even for all $1\le j,j'\le d$.

We shall say that the distribution of a positive semidefinite 
matrix valued measure $(G_{j,j'}(\cdot))$, $1\le j,j'\le d$, 
on ${\mathbb R}^\nu$ is moderately increasing at infinity if
\begin{equation}
\int (1+|x|)^{-r}G_{j,j}(\,dx)<\infty \quad\textrm{ for all }
1\le j\le d \textrm{ with some number } r>0.\label{4.1}
\end{equation}
}

\medskip\noindent
{\it Remark.} We can give, similarly to Lemma~2.3, a different 
characterization of positive semidefinite matrix valued, even 
measures on ${\mathbb R}^\nu$. Let us have some complex  
measures $G_{j,j'}$, $1\le j,j'\le d$, on the 
$\sigma$-algebra of the Borel measurable sets of ${\mathbb R}^\nu$ such 
that their restrictions to any cube $[-T,T]^\nu$, $T>0$, have 
finite total variation. Let us consider the matrix valued 
measure $(G_{j,j'}(A))$, $1\le j,j'\le d$ on ${\mathbb R}^\nu$ for all 
bounded, measurable sets $A\subset {\mathbb R}^\nu$. This matrix 
valued measure is positive semidefinite and even if and only
if it satisfies the following two conditions.

\medskip
\begin{description}
\item[{\rm(i.)}] The $d\times d$ matrix $(G_{j,j'}(A))$, 
$1\le j,j'\le d$, is Hermitian, positive semidefinite 
for all bounded, measurable sets $A\subset {\mathbb R}^\nu$.
\item[{\rm(ii.)}] $G_{j,j'}(-A)=\overline{G_{j,j'}(A)}$, for
all $1\le j,j'\le d$ and bounded, measurable sets 
$A\subset {\mathbb R}^\nu$.
\end{description}

\medskip
This statement has almost the same proof as Lemma~2.3.
The only difference in the proof is that now we have to
work with such vectors $v(x)=(v_1(x),\dots,v_d(x))$ whose 
coordinates $v_j(x)$ are continuous functions on ${\mathbb R}^\nu$ 
with bounded support, $1\le j\le d$. Let me also remark
that the following statement also follows from this proof.
If a matrix valued measure $(G_{j,j'}(A))$, $1\le j,j'\le d$,
on ${\mathbb R}^\nu$ satisfies the conditions in the definition of
positive semidefinite matrices with some $\sigma$-finite
measure $\mu$ on ${\mathbb R}^\nu$ with respect to which all 
complex measures $G_{j,j}$ are absolutely 
continuous, then it satisfies these conditions with any
 $\sigma$-finite measure $\mu$ on ${\mathbb R}^\nu$ with the 
same property. 

\medskip
Before constructing a large class of vector valued Gaussian 
stationary generalized random fields I recall an important 
property of the Fourier transform of the functions in the 
Schwartz spaces $\cal S$ and ${\cal S}^c$, (see e.g.~\cite{7}). 
Actually this property of the Schwartz spaces made useful 
their choice in the definition of generalized fields. 

The Fourier transform $f\to\tilde f$ is a bicontinuous map 
from ${\cal S}^c$ to~${\cal S}^c$. (This means that this 
transformation is invertible, and both the Fourier transform 
and its inverse are continuous maps from ${\cal S}^c$ to 
${\cal S}^c$.) (The restriction of the Fourier transform to 
the space ${\cal S}$ of real valued functions is a bicontinuous 
map from ${\cal S}$ to the subspace of ${\cal S}^c$ consisting 
of those functions $f\in{\cal S}^c$ for which
$f(-x)=\overline{f(x)}$ for all $x\in {\mathbb R}^\nu$.) 

Next I formulate the following result.

\medskip\noindent
{\bf Theorem 4.1 about the construction of vector valued Gaussian 
stationary generalized random fields with zero expectation.}
{\it Let $(G_{j,j'})$, $1\le j,j'\le d$, be a positive semidefinite 
matrix valued, even measure on ${\mathbb R}^\nu$ whose distribution is 
moderately increasing at infinity.

Then there exists a vector valued, Gaussian stationary 
generalized random field $(X_1(\varphi),\dots,X_d(\varphi))$,  
$\varphi\in{\cal S}$, such that $EX_j(\varphi)=0$ for all 
$\varphi\in{\cal S}$, and given two Shwartz functions 
$\varphi\in{\cal S}$ and $\psi\in{\cal S}$, the covariance
function $r_{j,j'}(\varphi,\psi)=EX_j(\varphi)X_{j'}(\psi)$ is 
given by the formula
\begin{equation}
r_{j,j'}(\varphi,\psi)=EX_j(\varphi)X_{j'}(\psi)
=\int\tilde\varphi(x)\,\bar{\!\tilde\psi}(x)G_{j,j'}(\,dx)  
\quad \textrm{for all } \varphi,\psi\in{\cal S}, \label{4.2}
\end{equation}
where $\,\tilde{}\,$ denotes Fourier transform, and $\,\bar{}\,$ is
complex conjugate. 

Formula~(\ref{4.2}) and the identity $EX_j(\varphi)=0$ for all 
$\varphi\in{\cal S}$ determine the distribution
of the vector valued, Gaussian stationary random field \hfill\break
$(X_1(\varphi),\dots,X_d(\varphi))$. 

Contrariwise, for 
all $1\le j,j'\le d$ the covariance function 
$EX_j(\varphi)X_{j'}(\psi)$, $\varphi,\psi\in{\cal S}$, 
determines the coordinate $G_{j,j'}$ of the positive 
semidefinite, even matrix $(G_{j,j'})$. $1\le j,j'\le d$, 
with moderately increasing distribution at infinity 
for which identity~(\ref{4.2}) holds.}

\medskip
Let me remark that the moderate decrease of the distribution of
the positive semidefinite matrix $(G_{j,j'})$, $1\le j,j'\le d$,
together with inequality~(\ref{3.2}) and the fast decrease of the 
functions $\varphi\in{\cal S}$ at infinity guarantee that the 
integral in~(\ref{4.2}) is convergent.

Condition~(\ref{4.1}) which we wrote in the definition of moderately
increasing positive semidefinite matrix valued measures appears
in the theory of generalized functions in a natural way. Such a
condition characterizes those measures which are generalized 
functions, i.e. continuous linear maps in the Schwartz space.

\medskip
In~\cite{11} we have proved with the help of some important results 
of Laurent Schwartz about generalized functions that in the 
case of scalar valued models, i.e. if $d=1$ the covariance 
function of every Gaussian stationary generalized random field 
with expectation zero agrees with the covariance function of 
a Gaussian stationary generalized random field constructed 
in the same way as we have done in Theorem~4.1. (In the case
$d=1$ the formulation of this result is simpler.) It seems 
very likely that a refinement of that argument would give 
the proof of an analogous statement in the general case. 
I did not investigate this question, because in the 
present paper we do not need such a result. 

\medskip\noindent
{\it Remark.}\/ Similarly to the case of vector valued 
stationary fields with discrete parameter we shall 
introduce the following terminology. If $(G_{j,j'})$, 
$1\le j,j'\le d$, is a positive semidefinite, matrix 
valued even measure with moderately increasing distribution 
at infinity, and there is a stationary generalized random field 
$(X_1(\varphi),\dots,X_d(\varphi))$, $\varphi\in{\cal S}$,
whose covariance function 
$$
r_{j,j'}(\varphi,\psi)=EX_j(\varphi)X_{j'}(\psi), \quad 
1\le j,j'\le d, \;\; \varphi,\psi\in{\cal S},
$$ 
satisfies relation~(\ref{4.2}) with this matrix valued 
measure $G$, then we call $G$ the matrix valued spectral 
measure of this covariance function $r_{j,j'}(\varphi,\psi)$. 
In general, we shall call a positive semidefinite matrix 
valued even measure on ${\mathbb R}^\nu$ with moderately increasing 
distribution at infinity a matrix valued spectral 
measure on ${\mathbb R}^\nu$. We have the right for such a 
terminology, because by Theorem~4.1 for any such matrix 
valued measure there exists a Gaussian stationary 
generalized random field such that this matrix valued 
measure is the matrix valued spectral measure of its 
covariance function. 

\medskip
Let me remark that the diagonal elements $G_{j,j}$ of the 
matrix valued spectral measure of the correlation function 
$r_{j,j'}(\varphi,\psi)$ of a vector valued stationary random 
field may have non finite measure on ${\mathbb R}^\nu$, they have to 
satisfy only relation~(\ref{4.1}). As a consequence, we can find 
a much richer class of matrix valued spectral measures by 
working with generalized random fields than by working only 
with classical stationary random fields. As we shall see 
also vector valued random spectral measures corresponding 
to these matrix valued spectral measures can be constructed.
Actually we discussed vector valued stationary generalized 
random fields in this paper in order to construct this 
larger class of matrix valued spectral and vector valued
random spectral measures. We are interested in them, 
because they appear in the limit theorems we shall prove. 

\medskip\noindent
{\it Proof of Theorem 4.1.}\/ Let us observe that the function 
$r_{j,j'}(\varphi,\psi)$ defined in~(\ref{4.2}) is real valued. This can
be seen  by applying the change of variables $x\to -x$ in this
integral and by exploiting that $G_{j,j'}(-A)=\overline{G_{j,j'}(A)}$,
and  $\tilde\varphi(-x)=\bar{\tilde\varphi}(x)$, 
$\tilde\psi(-x)=\bar{\tilde\psi}(x)$, since this calculation 
yields that $r_{j,j'}(\varphi,\psi)=\overline{r_{j,j'}(\varphi,\psi)}$. 
Let us also remark that 
$r_{j,j'}(\varphi,\psi)=r_{j',j}(\psi,\varphi)$, since by 
formula~(\ref{4.2})
and the property $G_{j,j'}(A)=\overline{G_{j',j}(A)}$ of the matrix 
$(G_{j,j'}(A))$, $1\le j,j'\le d$, for all measurable sets 
$A\subset {\mathbb R}^\nu$ we have
$r_{j,j'}(\varphi,\psi)=\overline{r_{j',j}(\psi,\varphi)}$, and
we know that both side of this identity is real valued.

First we show that for all positive integers $N$ and 
functions $\varphi_k\in{\cal S}$, $1\le k\le N$, there 
are some Gaussian random vectors
$(X_1(\varphi_k),\dots,X_d(\varphi_k))$, $1\le k\le N$,
with expectation zero and 
covariances $EX_j(\varphi_k)X_{j'}(\varphi_{k'})
=r_{j,j'}(\varphi_k,\varphi_{k'})$
for all $1\le j,j'\le d$, $1\le k,k'\le N$, 
on an appropriate probability space, where 
$r_{j,j'}(\varphi_k,\varphi_{k'})$ is defined at the 
right-hand side of formula~(\ref{4.2}) with our matrix
valued measure $(G_{j,j'})$, $1\le j,j'\le d$ and
with the choice $\varphi=\varphi_k$, $\psi=\varphi_{k'}$.

We prove this statement if we show that the matrix with 
elements 
$$
d_{(j,k),(j',k')}=r_{j,j'}(\varphi_k,\varphi_{k'}), \quad 
1\le j,j'\le d, \;\; 1\le k,k'\le N,
$$ 
is positive semidefinite. To prove this result take any vector 
$(a_{j,k},\;1\le j\le d, 1\le k\le N)$, and observe that
\begin{eqnarray*}
&&\sum_{j=1}^d\sum_{j'=1}^d\sum_{k=1}^N\sum_{k'=1}^N a_{j,k}\overline{a_{j',k'}}
r_{j,j'}(\varphi_k,\varphi_{k'}) \\
&&\qquad=\sum_{j=1}^d\sum_{j'=1}^d\sum_{k=1}^N\sum_{k'=1}^N 
\int (a_{j,k}\tilde\varphi_k(x))
(\overline{{a_{j',k'}\tilde\varphi}_{k'}(x)})
g_{j,j'}(x)\mu(\,dx)  \\
&&\qquad=\sum_{j=1}^d\sum_{j'=1}^d\int \psi_j(x)\overline{\psi_{j'}(x)}
g_{j,j'}(x)\mu(\,dx)
=\int \psi(x)g(x)\overline{\psi(x)}\,\mu(\,dx)\ge0,
\end{eqnarray*}
where $\psi_j(x)=\sum\limits_{k=1}^N a_{j,k}\tilde\varphi_k(x)$,
$1\le j\le d$, $\psi(x)=(\psi_1(x),\dots,\psi_d(x))$, and $g(x)$
denotes the matrix $(g_{j,j'}(x))$, $1\le j,j'\le d$. In this calculation
we applied formula~(\ref{4.2}), the representation 
$G_{j,j'}(\,dx)=g_{j,j'}(x)\mu(\,dx)$ and finally the fact that $g(x)$
is a semidefinite matrix for $\mu$ almost all~$x$. 

Then it follows from Kolmogorov's existence theorem for random 
processes with consistent finite distributions that there is 
a Gaussian random field 
$$
(X_1(\varphi),\dots,X_d(\varphi)), \quad \varphi\in{\cal S},
$$ 
with zero expectation such that
$EX_j(\varphi)X_{j'}(\psi)=r_{j,j'}(\varphi,\psi)$ for all 
functions $\varphi\in{\cal S}$, $(\psi\in{\cal S}$ and 
$1\le j,j'\le d$. Besides, the finite dimensional distributions 
of this random field are determined because of the Gaussian 
property. Next we show that this random field is a vector 
valued generalized random field.

Property~(a) of the vector valued generalized random fields
follows from the following calculation.
\begin{eqnarray*}
&&E[a_1X_j(\varphi)+a_2X_j(\psi)
-X_j(a_1\varphi+a_2\psi)]^2\\
&&\qquad =\int\left(a_1\tilde\varphi(x)+a_2\tilde\psi(x)
-(\widetilde{a_1\varphi+a_2\psi)}(x)\right) \\
&&\qquad\qquad\qquad \left(\overline{a_1\tilde\varphi(x)}
+\overline{a_2\tilde\psi(x)}
-\overline{(\widetilde{a_1\varphi+a_2\psi)}(x)}\right)
G_{j,j}(\,dx)=0
\end{eqnarray*}
by formula~(\ref{4.2}) for all real numbers $a_1$, $a_2$, 
$1\le j\le d$ and $\varphi,\psi\in{\cal S}$.

Property~(b) of the vector valued generalized random fields
also holds for this model. Actually it is proved in~\cite{11} that 
if $\varphi_n\to\varphi$ in the topology of the space 
${\cal S}$, then 
$E[X_j(\varphi_n)-X_j(\varphi)]^2
=\int|\tilde\varphi_n(x)-\tilde\varphi(x)|^2 G_{j,j}(\,dx)\to0$ 
as $n\to\infty$, hence property~(b) also holds. (The proof is
not difficult. It exploits that for a sequence of functions
$\varphi_n\in{\cal S}^c$, $n=0,1,2,\dots$, $\varphi_n\to\varphi_0$
as $n\to\infty$ in the topology of ${\cal S}^c$ if and only if
$\tilde \varphi_n\to\tilde\varphi_0$ in the same topology.  
Besides, the measure $G_{j,j}$ satisfies inequality~(\ref{4.1}).)

It is also clear that the Gaussian random field constructed in 
such a way is stationary.

It remained to show that the covariance function 
$r_{j,j'}(\varphi,\psi)=EX_j(\varphi)X_{j'}(\psi)$
determines the complex measure $G_{j,j'}$. To show 
this we have to observe that inequality~(\ref{3.2}) holds also in 
this case, hence the Schwarz inequality implies that
$$
\int (1+|x|)^{-r}|g_{j,j'}(x)|\mu(\,dx)<\infty \quad\textrm{ for all }
1\le j,j'\le d 
$$
for a positive semidefinite matrix valued measure with moderately 
increasing distribution, i.e. this inequality holds not only for
$j=j'$. Then it follows from the standard theory of Schwartz
spaces that the class of Schwartz functions is sufficiently rich
to guarantee that the function $r_{j,j'}(\varphi,\psi)$ determines
the complex measure $G_{j,j'}$. Theorem~4.1 is proved.

\medskip
Next we construct a vector valued random spectral measure 
corresponding to a matrix valued spectral measure $(G_{j,j'})$,
$1\le j,j'\le d$, on~${\mathbb R}^\nu$. We argue similarly to Section~3, 
where the vector valued random spectral measures corresponding
to matrix valued spectral measures on $[-\pi,\pi)^\nu$ were 
considered. In the construction we shall also refer to some 
results in~\cite{11}.

Let us have a vector valued, Gaussian stationary generalized
random field $X=(X_1(\varphi),\dots,X_d(\varphi))$, 
$\varphi\in{\cal S}$, $1\le j\le d$, with a matrix 
valued spectral measure $(G_{j,j'})$, $1\le j,j'\le d$.
First we define for all $1\le j\le d$ some (complex) 
Hilbert spaces ${\cal K}^c_{1,j}$, ${\cal H}^c_{1,j}$ and 
a norm preserving, invertible linear transformation 
$T_j$ between them in the following way. ${\cal K}^c_{1,j}$ 
consists of those complex valued functions $u(x)$ on 
${\mathbb R}^\nu$ for which $\int |u(x)|^2G_{j,j}(\,dx)<\infty$ 
with the scalar product 
$\langle u(x),v(x)\rangle=\int u(x)\overline{v(x)}G_{j,j}(\,dx)$.
To define the Hilbert space ${\cal H}^c_{1,j}$ let us first 
introduce the Hilbert space ${\cal H}={\cal H}^c$ of (complex 
valued) random variables with finite second moment on the 
probability space $(\Omega,\cal A,P)$ where our  stationary 
generalized random field is defined. We define the Hilbert
space ${\cal H}^c$ in the space consisting of these random
variables with the usual scalar product 
$\langle\xi,\eta\rangle=E\xi\bar\eta$ in ${\cal H}^c$. The 
Hilbert space ${\cal H}^c_{1,j}$ is defined  as the closure 
of the linear subspace of ${\cal H}^c$ consisting of the 
complex valued random variables $X_j(\varphi)+iX_j(\psi)$, 
$\varphi,\psi\in{\cal S}$. 

First we define the operator $T_j$ for functions of the 
form $\widetilde{\varphi+i\psi}$, 
$\varphi,\psi\in{\cal S}$. We define it by the formula
\begin{equation}
T_j(\widetilde{\varphi+i\psi)}=X_j(\varphi)+iX_j(\psi),
\quad \varphi,\,\psi\in{\cal S}. \label{4.3}
\end{equation}
Some calculation which was actually carried out in~\cite{11} 
shows that the set of functions
$\widetilde{\varphi+i\psi}$, $\varphi,\psi\in{\cal S}$, is 
dense in ${\cal K}^c_{1,j}$, and the transformation $T_j$, 
defined in~(\ref{4.3}) can be extended to a norm preserving, 
invertible linear transformation from ${\cal K}^c_{1,j}$ 
to ${\cal H}^c_{1,j}$. (In the calculation leading to this 
statement we apply formula~(\ref{4.2}) with the choice $j'=j$.)

Then we can define the random spectral measure $Z_{G,j}(A)$, 
similarly to the case discussed in Section~3, by the formula 
$Z_{G,j}(A)=T_j{\mathbb I}_A(\cdot))$ for all bounded 
measurable sets $A\subset {\mathbb R}^\nu$. To determine 
the joint distribution of the spectral measures $Z_{G,j}$ 
we make the following version of the corresponding 
argument in Section~3.

We define the following two Hilbert spaces ${\cal K}^c_1$
and ${\cal H}^c_1$ together with a norm preserving linear
transformation $T$ between them.

The elements of the Hilbert space ${\cal K}^c_1$ are 
the vectors $u=(u_1(x),\dots,u_d(x))$ with 
$u_j(x)\in{\cal K}^c_{1,j}$, $1\le j\le d$. We define
the scalar product on ${\cal K}^c_1$ with the 
help of the following positive semidefinite bilinear 
form $\langle \cdot,\cdot\rangle_0$. If 
$u(x)=(u_1(x),\dots,u_d(x))\in{\cal K}^c_1$  and 
$v(x)=(v_1(x),\dots,v_d(x))\in{\cal K}^c_1$, then
\begin{eqnarray*}
\langle u(x), v(x)\rangle_0
\!&=& \! \sum_{j=1}^d\sum_{j'=1}^d\int u_j(x)
\overline{ v_{j'}(x)}G_{j,j'}(\,dx) \\
\! &=& \! \sum_{j=1}^d\sum_{j'=1}^d\int g_{j,j'}(x)u_j(x)
\overline{ v_{j'}(x)}\mu(\,dx) 
=\int u(x) g(x) v(x)^*\mu(\,dx) 
\end{eqnarray*}
with the matrix $g(x)=(g_{j,j'}(x))$, $1\le j,j'\le d$, where 
$v^*(x)$ denotes the column vector whose elements are
the functions $\overline {v_{j'}(x)}$, $1\le j'\le d$. 
Actually here we simply copied the corresponding
definition in Section~3 for the discrete time model,
and we can also prove that ${\cal K}^c_1$ is a Hilbert
space with the scalar $\langle\cdot,\cdot\rangle_0$ in the
same way as it was done in Section~3. 

The construction ${\cal H}^c_1$ and the proof of its
properties is again a simple copying of argument made
in Section~3. The elements of ${\cal H}^c_1$ are the
vectors $\xi=(\xi_1,\dots,\xi_d)$, where 
$\xi_j\in{\cal H}^c_{1,j}$, $1\le j\le d$, and we 
define the norm on it by means of the scalar product 
$\langle\xi,\eta\rangle_1=E\left(\sum_{j=1}^d\xi_j\right)
\overline{\left(\sum_{j=1}^d\eta_j\right)}$ for
$\xi=(\xi_1,\dots,\xi_d)\in{\cal H}^c_1$ and
$\eta=(\eta_1,\dots,\eta_d)\in{\cal H}^c_1$. We identify
two elements $\xi\in{\cal H}^c_1$ and $\eta\in{\cal H}^c_1$
if $\|\xi-\eta\|_1=0$. Then the argument of Section~3
yields that  ${\cal H}^c_1$ is a Hilbert space with the
scalar product $\langle\cdot,\cdot\rangle_1$.

We define the operator $T$ from ${\cal K}^c_1$ to ${\cal H}^c_1$ 
again in the same way as in Section~3. We define it by the
formula
$$
Tu=T(u_1,\dots,u_d)=(T_1u_1,\dots,T_du_d)
$$  
for 
$u=(u_1,\dots,u_d)$, $u_j\in{\cal K}^c_{1,j}$, with the help 
of the already defined operators $T_j$, $1\le j\le d$. We 
want to show that it is a norm preserving and invertible 
transformation from ${\cal K}^c_1$ to ${\cal H}^c_1$. Here
again we apply a similar, but sightly different argument 
from that in Section~3. We exploit that if we take the 
class of vectors 
$$
W=\{w=(u_1+iv_1,\dots,u_d+iv_d)\colon\; u_j\in{\cal S},\; 
v_j\in{\cal S} \textrm{ for all } 1\le j\le d\}
$$
then the class of vectors 
$$
\tilde W=\{(\widetilde{u_1+iv_1},\dots,\widetilde{u_d+iv_d})\colon\;
(u_1+iv_1,\dots, u_d+iv_d)\in W\}
$$ 
is an everywhere dense subspace of ${\cal K}^c_1$. and 
the class of vectors
$$
W(X)=\{((X_1(u_1+iv_1),\dots,X_d(u_d+iv_d)) \colon\; 
(u_1+iv_1,\dots,u_d+iv_d)\in W\}
$$ 
is an everywhere dense subspace of ${\cal H}^c_1$. (Here 
again the sign $\tilde{\ }$ denotes Fourier transform.)

Take two vectors 
$(u_{1,1}+iv_{1,1},\dots, u_{d,1}+iv_{d,1})\in W$ 
and $(u_{1,2}+iv_{1,2},\dots, u_{d,2}+iv_{d,2})\in W$.
The desired property of the operator~$T$ will follow from 
the following calculation.
\begin{eqnarray*}
&&\langle (\widetilde{u_{1,1}+iv_{1,1}},\dots,\widetilde{u_{d,1}+v_{d,1}}),
(\widetilde{u_{1,2}+iv_{1,2}},\dots,\widetilde{u_{d,2}+v_{d,2}})\rangle_0 \\
&&\qquad=\sum_{j=1}^d\sum_{j'=1}^d\int 
\widetilde{(u_{j,1}(x)+iv_{j,1}(x))} 
\overline{\widetilde{(u_{j',2}(x)+iv_{j',2}(x))}}G_{j,j'}(\,dx)\\
&&\qquad
=\sum_{j=1}^d\sum_{j'=1}^d E[X_j(u_{j,1})+iX_j(v_{j,1})]
[X_{j'}(u_{j',2})-iX_j(u_{j',2})] \\
&&\qquad=\langle (X_1(u_{1,1})+iX_1(v_{1,1}),\dots, 
X_d(u_{d,1})+iX_d(v_{d,1})), \\
&&\qquad\qquad\qquad 
(X_1(u_{1,2})+iX_1(v_{1,2}),\dots, X_d(u_{d,2})+iX_d(v_{d,2}))\rangle_1,
\end{eqnarray*}
i.e.
\begin{eqnarray*}
&&\langle (\widetilde{u_{1,1}+iv_{1,1}},\dots,\widetilde{u_{d,1}+v_{d,1}}),
(\widetilde{u_{1,2}+iv_{1,2}},\dots,\widetilde{u_{d,2}+v_{d,2}})\rangle_0 \\
&&\qquad=\langle (T_1(u_{1,1}+iv_{1,1}),\dots, 
T_d(u_{d,1}+iv_{d,1})), \\
&& \qquad \qquad \qquad \qquad
(T_1(u_{1,2}+iv_{1,2})),\dots, T_d(u_{d,2}+iv_{d,2}))\rangle_1.
\end{eqnarray*}
This means that the operator $T$ maps the everywhere dense
subspace $\tilde W$ of ${\cal K}^c_1$ to the everywhere 
dense subspace $W(X)$ of ${\cal H}^c_1$ in a norm preserving
form. This implies that $T$ is a norm preserving, invertible 
transformation from ${\cal K}^c_1$ to ${\cal H}^c_1$.

Now we turn to the definition of the vector valued random
spectral measures corresponding to a matrix valued spectral
measure on ${\mathbb R}^\nu$. 

Let a vector valued, Gaussian stationary generalized 
random field 
$$
X(\varphi)=(X_1(\varphi),\dots,X_d(\varphi)), 
\quad\varphi\in{\cal S},
$$ 
be given with a matrix valued spectral measure 
$(G_{j,j'})$, $1\le j,j'\le d$, on ${\mathbb R}^\nu$. (We 
take such generalized, stationary random fields which 
were constructed in Theorem~4.1.) Let us consider the 
operators $T_j$, $1\le j\le d$, and $T$ constructed
above with the help of these quantities.  We define, 
similarly to the case of Gaussian stationary random 
fields with discrete parameters discussed in Section~3 
the random variables  $Z_{G,j}(A)=T_j({\mathbb I}_A(x))$ 
for all $1\le j\le d$ and bounded, measurable sets 
$A\subset {\mathbb R}^\nu$. (These functions 
${\mathbb I}_A(\cdot)$ are clearly elements of the 
Hilbert space ${\cal K}^c_{1,j}$ for all $\le j\le d$). 
It can be proved with the help of the properties of 
the operator~$T$ that these random functions satisfy 
properties (i)--(v) formulated in the definition of 
random spectral measures on the torus, considered in 
Section~3. The argument applied in Section~3 holds 
also in in this case. In particular, property~(v) 
can be proved with the help of property~(v$'$). 
Property~(v$'$) can be proved with some work, and 
actually this was done in~\cite{11}. We prove~(v$'$) 
by checking it first for functions $u\in{\cal S}^c$.

The above result makes natural the following definition of 
vector valued random spectral measures corresponding to a 
matrix valued spectral measure on ${\mathbb R}^\nu$. This is very
similar to the definition of vector valued random spectral
measures on the torus.

\medskip\noindent
{\bf Definition of vector valued random spectral measures on 
${\mathbb R}^\nu$.} {\it Let $G=(G_{j,j'})$, $1\le j,j'\le d$, be 
a matrix valued spectral measure on ${\mathbb R}^\nu$. We call 
a set of complex valued random variables $Z_{G,j}(A)$
depending on pairs $(j,A)$, where $1\le j\le d$,  
$A\in{\cal A}$, and ${\cal A}$ is the algebra  
$$
{\cal A}=\{ A\colon\; A 
\textrm{ is a bounded Borel measurable set in } {\mathbb R}^\nu\},
$$ 
a $d$-dimensional vector valued random spectral measure 
corresponding to the matrix valued spectral measure~$G$ 
on ${\mathbb R}^\nu$ if this set of random variables 
$Z_{G,j}(A)$, $1\le j\le d$, $A\in{\cal A}$, satisfies 
properties~(i)--(v) introduced in Section~3 in the 
definition of vector valued random spectral measures on 
the torus. Given a fixed index $1\le j\le d$, we call the
set of random variables $Z_{G,j}(A)$, $A\in{\cal A}$,
with this index $j$ the $j$-th coordinate of this matrix
valued spectral measure, and we denote it by~$Z_{G,j}$.
We denote a $d$-dimensional vector valued random spectral 
measure corresponding to the matrix valued spectral 
measure~$G$ by $Z_G=(Z_{G,1},\dots,Z_{G,d})$. }

\medskip
We have shown with the help of the arguments applied in
Section~3 that for any $d$-dimensional matrix valued spectral 
measure on ${\mathbb R}^\nu$ there exists a $d$-dimensional vector 
valued random spectral measure corresponding to it. 
%We can define the random integral with respect to it in the same
%way as it was done for vector valued random spectral measures
%on the torus in Section~3.

We can define the random integral $\int f(x)Z_{G,j}(\,dx)$ of 
the functions $f\in{\cal K}^c_{1,j}$ with respect to the random 
spectral measure $Z_{G,j}$, $1\le j\le d$, corresponding to 
the matrix valued spectral measure $(G_{j,j'})$, $1\le j,j'\le d$, 
of a Gaussian stationary generalized field in the same way as we
defined these random  integrals with respect to random spectral
measures corresponding to a spectral measures on the torus
$[-\pi,\pi)^\nu$ in Section~3. First we define these integrals for 
elementary functions which are defined in the same way as
it was done in Section~3. Then following the calculation
of that section we can define these integrals for a general
function $f\in{\cal K}^c_{1,j}$, and it can be seen that
formulas (\ref{3.7}), (\ref{3.8}) and (\ref{3.9}) remain 
valid for them. In particular, the random integrals 
$\int \tilde \varphi(x)Z_{G,j}(\,dx)$ are (meaningful and) real 
valued random variables for all $\varphi\in{\cal S}$, and
$$
E \left(\int \tilde \varphi(x)Z_{G,j}(\,dx)
 \int\bar{\tilde \psi}(x)Z_{G,{j'}}(\,dx)\right)
=\int \tilde\varphi(x)\bar{\tilde \psi}(x) G_{j,j'}(\,dx)
$$
for all $\varphi,\psi\in {\cal S}$ and $1\le j,j'\le d$. This 
identity together with relation~(\ref{3.7}) and the fact that the 
above considered random integrals are linear operators imply 
that the set of random variables
\begin{equation}
X_j(\varphi)=\int\tilde\varphi (x) Z_{G,j}(\,dx), 
\quad \varphi\in{\cal S}, \;\; 1\le j\le d, \label{4.4}
\end{equation}
constitute a vector valued Gaussian, stationary generalized
random field with spectral measure $(G_{j,j'})$, $1\le j,j'\le d$.

This implies that the natural version of Theorem~3.1 remains
valid if we consider a matrix valued spectral measure~$(G_{j,j'})$,
$1\le j,j'\le d$, on ${\mathbb R}^\nu$. Then there exists a random spectral
measure $Z_G=(Z_{G,1},\dots,Z_{G,d})$ corresponding to it, and
we have defined the random integrals $\int u(x)Z_{G,j}(\,dx)$,
$1\le j\le d$, with respect to it for all $u\in{\cal K}^c_{1,j}$. 
The class of random variables, $X_j(\varphi)$, 
$\varphi\in{\cal S}$, $1\le j\le d$, defined in~(\ref{4.4})
constitute a vector valued, Gaussian stationary generalized 
random field with matrix valued spectral measure $(G_{j,j'})$, 
$1\le j,j'\le d$. Moreover, if a $d$-dimensional vector 
valued Gaussian stationary random field is given with 
spectral measure $(G_{j,j'})$, $1\le j,j'\le d$, then we can 
consider  the random spectral measure $(Z_{G,1},\dots,Z_{G,d})$ 
constructed in this section with the help of this random 
field. This random spectral measure has the property that 
the random field given by the random integrals defined 
in formula~(\ref{4.4}) with their help agrees with the original 
vector valued Gaussian stationary generalized random field.

We can formulate a natural version of Lemma~3.2 where we
consider a matrix valued spectral measure $(G_{j,j'})$, 
$1\le j,j'\le d$, on ${\mathbb R}^\nu$ instead of a matrix valued 
spectral measure on the torus $[-\pi,\pi)^\nu$. In this
version of Lemma~3.2 we define ${\cal K}_{1,j}$ as
$$
{\cal K}_{1,j}=\left\{u\colon \int |u(x)|^2G_{j,j}(\,dx)<\infty, \quad 
u(-x)=\overline{u(x)}\textrm{ for all } x\in {\mathbb R}^\nu\right\},
$$
with the scalar product
$\langle u,v\rangle=\int u(x)\overline{v(x)}G_{j,j}(\,dx)$, 
$u,v\in{\cal K}_{1,j}$, and ${\cal H}_{1,j}$ as the closure of 
the linear space consisting of the random variables $X_j(\varphi)$, 
$\varphi\in{\cal S}$, in the Hilbert space ${\cal H}$. 
This version of Lemma~3.2 states that ${\cal K}_{1,j}$ and 
${\cal H}_{1,j}$ are real Hilbert spaces, and 
$T_j(u)=\int u(x)Z_{G,j}(\,dx)$ is a norm preserving and
invertible transformation from ${\cal K}_{1,j}$ to 
${\cal H}_{1,j}$.

The proof of this version of Lemma~3.2 is very similar to the proof 
of the original lemma. The main difference is that now we show that
the class of functions $\tilde\varphi$ with $\varphi\in{\cal S}$ is
a dense linear subspace of ${\cal K}_{1,j}$, and the transformation
$T_j(\tilde{\varphi})=\int \tilde{\varphi}(x)Z_{G,j}(\,dx)=X_j(\varphi)$,
$\varphi\in{\cal S}$, is a norm preserving transformation from 
an everywhere dense subspace of ${\cal K}_{1,j}$ to an everywhere
dense subspace of ${\cal H}_{1,j}$.

The natural version of Lemma~3.3 also holds. It states that a
matrix valued spectral measure $(G_{j,j'})$, $1\le j,j'\le d$, 
on ${\mathbb R}^\nu$ determines the distribution of a vector 
valued random spectral measure $Z_{G,j}$, $1\le j\le d$, 
corresponding to it. The proof of this version is the same as 
the proof of the original lemma. The only difference is that 
now we consider the random spectral measure $Z_{G,j}(A)$ for 
all measurable, bounded sets $A\subset {\mathbb R}^\nu$.

Finally I would remark that property (vi) of the random
spectral measures also remains valid for this new class of 
random spectral measures, because its proof applies only 
properties (i)--(v) of random spectral measures.

\section{Multiple Wiener--It\^o integrals with respect to vector 
valued random spectral measures}

Next we want to rewrite the random variables with finite second
moments which are measurable with respect the $\sigma$-algebra
generated by the elements of a vector valued Gaussian stationary 
random field  in an appropriate form which enables us  to rewrite 
also the random sums defined in~(\ref{1.1}) in a form that 
helps in the study of their limit behaviour. In the scalar 
valued case, i.e. when $d=1$ we could do this with the help of 
the introduction of multiple Wiener--It\^o integrals. We could 
rewrite with their help the random sums~(\ref{1.1}) in a form 
that provided great help in the study of the limit theorems we 
were interested in. Next we show that a similar method can be 
applied also in the case of vector valued Gaussian stationary
fields. To do this first we have to define the multiple 
Wiener--It\^o integrals also in the vector valued case. We 
start the definition of multiple  Wiener--it\^o integrals in 
this case with the introduction of the following notation.

\medskip
Let $X(p)=(X_1(p),\dots,X_d(p))$, $EX(p)=0$, 
$p\in{\mathbb Z}^\nu$, be a vector valued 
stationary Gaussian random field with some matrix 
valued spectral measure 
$G=(G_{j,j'})$, $1\le j,j'\le d$. Let
$Z_G=(Z_{G,1},\dots,Z_{G,d})$ be a
vector valued random spectral measure 
corresponding to it which is chosen in such
a way 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 us consider 
the (real) Hilbert space ${\cal H}$ of square integrable 
random variables measurable with respect to the 
$\sigma$-algebra generated by the random vectors $X(p)$, 
$p\in{\mathbb Z}^\nu$. More generally, let us consider a 
(possibly generalized) matrix valued spectral measure 
$G=(G_{j,j'})$, $1\le j,j'\le d$, and a vector valued random 
spectral measure $Z_G=(Z_{G,1},\dots,Z_{G,d})$ corresponding 
to it, where the matrix valued spectral measures $G_{j,j'}$ 
and vector valued random spectral measures $Z_{G,j}$ are 
defined  either on the torus $[-\pi,\pi)^\nu$ or on ${\mathbb R}^\nu$, 
and consider the (real) Hilbert space ${\cal H}$ of the 
square integrable (real valued) random variables, 
measurable with respect to the $\sigma$-algebra generated 
by the random variables of the vector valued random 
spectral measures $Z_G$ with the usual scalar product 
in this space. We would like to write the elements of 
the Hilbert space ${\cal H}$ in the form of a sum of 
multiple Wiener--It\^o integrals with respect to the 
vector valued random spectral measure $Z_G$. I shall 
construct these Wiener--It\^o integrals in this section, 
and I prove some of their important properties. 

As a discussion in Section~7 will show we cannot write 
all elements of ${\cal H}$ in the form of a sum of 
Wiener--It\^o integrals, but we can do this for the 
elements of an everywhere dense subspace of ${\cal H}$.
In particular, if we consider finitely many random variables
$X_j(p)$, $1\le j\le d$, $p\in{\mathbb Z}^\nu$ of a discrete
or $X_j(\varphi)$, $1\le j\le d$, $\varphi\in{\cal S}^\nu$,
of a generalized vector valued stationary Gaussian 
random field, then all polynomials of these random
variables can be written as the sum of Wiener--It\^o
integrals. Such a result will be sufficient for our
purposes. In the subsequent discussion I impose a 
technical condition about the properties of the matrix 
valued spectral measure $G=(G_{j,j'})$ I shall be working 
with. I assume that it is non-atomic. More precisely, 
I assume that we are working with such a dominating 
measure $\mu$ for the coordinates of the matrix valued 
spectral measures $G_{j,j'}$ for which 
$\mu(\{x\})=0$ for all $x\in\ {\mathbb R}^\nu$.

First I define for all $n=1,2,\dots$ and $1\le j_s\le d$ 
for the indices $1\le s\le n$ the $n$-fold multiple 
Wiener--It\^o integral
$$
I_n(f|j_1,\dots,j_n)=\int f(x_1,\dots,x_n)
Z_{G,{j_1}}(\,dx_1)\dots Z_{G,{j_n}}(d\,x_n)
$$
with respect to the coordinates of a vector valued random 
spectral measure $Z_G=(Z_{G,1},\dots,Z_{G,d})$, corresponding 
to a matrix valued spectral measure $G=(G_{j,j'})$,
$1\le j,j'\le d$. We shall define these Wiener--It\^o 
integrals functions with kernel functions 
$f\in{\cal K}_{n,j_1,\dots,j_n}$ in a (real) Hilbert space 
${\cal K}_{n,j_1,\dots,j_n}
={\cal K}_{n,j_1,\dots,j_n}(G_{j_1,j_1},\dots,G_{j_n,j_n})$ 
defined below.

We define ${\cal K}_{n,j_1,\dots,j_n}
={\cal K}_{n,j_1,\dots,j_n}(G_{j_1,j_1}.\dots,G_{j_n,j_n})$ as the 
Hilbert space consisting of those complex valued functions
$f(x_1,\dots,x_n)$ on ${\mathbb R}^{n\nu}$ which satisfy the following 
relations~(a) and~(b):

\medskip
\begin{description}
\item[{\rm(a)}] $f(-x_1,\dots,-x_n)=\overline{f(x_1,\dots,x_n)}$ 
for all $(x_1,\dots,x_n)\in {\mathbb R}^{n\nu}$,
\item[{\rm(b)}]
$\|f\|^2=\int|f(x_1,\dots,x_n)|^2
G_{j_1,j_1}(\,dx_1)\dots G_{j_n,j_n}(\,dx_n)<\infty$.
\end{description}

\medskip\noindent
We define the scalar product in ${\cal K}_{n,j_1,\dots,j_n}$ in the 
following way. If $f,\,g\in{\cal K}_{n,j_1,\dots,j_n}$, then
\begin{eqnarray*}
\langle f,g\rangle&=&\int f(x_1,\dots,x_n)\overline{g(x_1,\dots,x_n)}
G_{j_1,j_1}(\,dx_1)\dots G_{j_n,j_n}(\,dx_n)\\
&=&\int f(x_1,\dots,x_n)g(-x_1,\dots,-x_n)
G_{j_1,j_1}(\,dx_1)\dots G_{j_n,j_n}(\,dx_n).
\end{eqnarray*}

Because of the symmetry $G_{j_s,j_s}(A)=G_{j_s,j_s}(-A)$ of the 
spectral measure
$\langle f,g\rangle=\overline{\langle f,g\rangle}$, i.e. the 
scalar product $\langle f,g\rangle$ is a real
number for all $f,\,g\in{\cal K}_{n,j_1,\dots,j_n}$. This means that
${\cal K}_{n,j_1,\dots,j_n}$ is a real Hilbert space, as I claimed.
We also define the real Hilbert space ${\cal K}_0$ for $n=0$ as
the space of real constants with the norm $\|c\|=|c|$. 

\medskip\noindent
{\it Remark.}\/ In the case $n=1$ the above defined real Hilbert 
space ${\cal K}_{1,j}$ agrees with the real Hilbert 
space~${\cal K}_{1,j}$ introduced in Lemma~3.2.

\medskip
Similarly to the scalar valued case, first we introduce so-called
simple functions and define the multiple integrals for them. We
prove some properties of this integral which enable us to extend
its definition by means of an $L_2$ extension for all functions 
$f\in{\cal K}_{j_1,\dots,j_n}$. We define the class of simple 
functions together with the notion of regular systems.

\medskip\noindent
{\bf Definition of regular systems and the class of simple functions.}
{\it Let 
$$
{\cal D}=\{\Delta_k,\;k=\pm1,\pm2,\dots,\pm N\}
$$ 
be a finite collection of bounded, measurable sets in ${\mathbb R}^\nu$  
indexed by the integers $\pm1$,\dots, $\pm N$ with some positive
integer~$N$. We say that ${\cal D}$ is a regular system if
$\Delta_k=-\Delta_{-k}$, and $\Delta_k\cap\Delta_l=\emptyset$
if $k\neq l$ for all $k,l=\pm1,\pm2,\dots,\pm N$. A function
$f\in{\cal K}_{n,j_1,\dots,j_n}$ is adapted to this system 
${\cal D}$ if $f(x_1,\dots,x_n)$ is constant on the sets
$\Delta_{k_1}\times\Delta_{k_2}\times\cdots\times\Delta_{k_n}$, \
$k_l=\pm1,\dots,\pm N$, $l=1,2,\dots,n$,  it vanishes outside these
sets, and it also vanishes on those sets of the above form for which 
$k_l=\pm k_{l'}$ for some $l\neq l'$. 

A function $f\in{\cal K}_{n,j_1,\dots,j_n}$ is in the class
$\hat{\cal K}_{n,j_1,\dots,j_n}$ of simple functions if it is adapted to
some regular system ${\cal D}=\{\Delta_k,\;k=\pm1,\dots,\pm N\}$.} 

\medskip\noindent
{\bf Definition of Wiener--It\^o integrals of simple functions.}
{\it Let a simple function $f\in \hat{\cal K}_{n,j_1,\dots,j_n}$ 
be adapted  to some regular system 
$$
{\cal D}=\{\Delta_k,\;k=\pm1,\dots,\pm N\}.
$$ 
Its $n$-fold Wiener--It\^o integral with respect to 
$Z_G=(Z_{G,1},\dots,Z_{G,d})$ with parameters 
$j_1,\dots,j_n$, $1\le j_k\le d$ for all 
$1\le k\le n$, is defined as
\begin{eqnarray}
&&\int f(x_1,\dots,x_n)Z_{G,{j_1}}(\,dx_1)\dots Z_{G,{j_n}}(\,dx_n) 
\label{5.1} \\
&&\qquad =I_n(f|j_1,\dots,j_n) \nonumber  \\
&&\qquad=\sum_{\substack{k_l=\pm1,\dots,\pm N\\l=1,2,\dots,n}}
f(u_{k_1},\dots,u_{k_n})
Z_{G,{j_1}}(\Delta_{k_1})\cdots Z_{G,{j_n}}(\Delta_{k_n}),
\nonumber
 \end{eqnarray}
where $u_k\in\Delta_k$, $k=\pm1,\dots,\pm N$.} 

\medskip\noindent
Although the regular system ${\cal D}$ to which $f$ is adapted is 
not uniquely determined (e.g. the elements of ${\cal D}$ can be 
divided to smaller sets), the integral defined in~(\ref{5.1}) is 
meaningful, i.e. its value does not depend on the choice of 
${\cal D}$. This can be proved with the help of property~(iv) 
of vector valued random spectral measures defined in Section~3 
in the same way as it was done in the scalar valued 
case in~\cite{11}. (Let me also remark that here I defined the random 
integral $I_n(f|j_1,\dots,j_n)$ with a normalization different 
from the normalization of the corresponding expression $I_G(f)$ 
introduced in~\cite{11}. Here I omitted the norming term $\frac1{n!}$.)

\medskip
Because of the definition of simple functions the sum in~(\ref{5.1})
does not change if we allow in it summation only for such sequences
$k_1,\dots,k_n$ for which $k_l\neq\pm k_{l'}$ if $l\neq l'$. This
fact will be exploited in the subsequent considerations.

\medskip
Next I formulate some important properties about the 
Wiener--It\^o integrals of simple functions. Later we shall 
see that these properties remain valid in the general case.

\begin{equation}
I_n(f|j_1,\dots,j_n) 
\textrm{ is a real valued random variable for all } 
f\in\hat{\cal K}_{n,j_1,\dots,j_n}. \label{5.2}
\end{equation}
Indeed, $I_n(f|j_1,\dots,j_n)=\overline{I_n(f|j_1,\dots,j_n))}$
by Property~(a) of the functions in ${\cal K}_{n,j_1,\dots,j_n}$ 
and property~(v) of the random spectral measures defined in 
Section~3, hence~(\ref{5.2}) holds. It is also clear that  
$\hat{\cal K}_{n,j_1,\dots,j_n}$ is a linear space, and the mapping
$f\to I_n(f|j_1,\dots,j_n)$ is a linear transformation on it.

The relation
\begin{equation}
EI_n(f|j_1,\dots,j_n)=0 \quad 
\textrm{for }f\in\hat{\cal K}_{n,j_1,\dots,j_k} \quad
\textrm { if } n\neq0 \label{5.3}
\end{equation}
also holds. (In the non-zero terms of the sum in~(\ref{5.1}) we have 
the product of independent random variables with expectation 
zero by property~(vi) of the random spectral measures described
also in Section~3.) Next I express the covariance 
between random variables of the form  $I_n(f|j_1,\dots,j_n)$. 
To do this first I introduce the following notation. Let 
$\Pi(n)$ denote the set of all permutations of the set 
$\{1,\dots,n\}$, and let $\pi=(\pi(1),\dots,\pi(n))$ denote 
one of its element. 

Let us have a positive integer $n\ge1$, and two sequences
$j_1,\dots,j_n$ and $j'_1,\dots,j'_n$, $1\le j_s,j'_s\le d$ for all 
$1\le s\le d$. Let $f\in\hat{\cal K}_{n,j_1,\dots,j_n}$ and 
$h\in\hat{\cal K}_{n,j'_1,\dots,j'_n}$. I shall show that
\begin{eqnarray}
&&EI_n(f|j_1,\dots,j_n)I_n(h(|j'_1,\dots,j'_n) \label{5.4} \\
&&\qquad =\sum_{\pi\in\Pi(n)} \int f(x_1,\dots x_n) 
\overline{h(x_{\pi(1)},\dots,x_{\pi(n)})} \nonumber \\
&&\qquad\qquad\qquad 
G_{j_1,j'_{\pi^{-1}(1)}}(\,dx_1)\dots G_{j_n,j'_{\pi^{-1}(n)}}(\,dx_n). 
\nonumber
\end{eqnarray}
On the other hand, if $n\neq n'$, and 
$f\in\hat{\cal K}_{n,j_1,\dots,j_n}$,
$h\in\hat{\cal K}_{n',j'_1,\dots,j'_{n'}}$, then
\begin{equation}
EI_n(f|j_1,\dots,j_n)I_{n'}(h(|j'_1,\dots,j'_{n'})=0. \label{5.5}
\end{equation}

Next I show the following inequality with the help of formula~(\ref{5.4}).
\begin{eqnarray}
E|I_n(f|j_1,\dots,j_n)|^2 &\le& n! \int |f(x_1,\dots x_n)|^2
G_{j_1,j_1}(\,dx_1)\dots G_{j_n,j_n}(\,dx_n) \nonumber \\
&=&n!\|f_{n,j_1,\dots,j_n}\|^2 \label{5.6}
\end{eqnarray}
for all $f\in\hat{\cal K}_{n,j_1,\dots,j_n}$.

Indeed we get by applying (\ref{5.4}) for 
$f=h\in\hat{\cal K}_{n,j_1,\dots,j_n}$
together with relation~(\ref{3.2}) that
\begin{eqnarray}
&&\!\!\!\!\!\! E|I_n(f|j_1,\dots,j_n)|^2 
 \le\sum_{\pi\in\Pi(n)} \int |f(x_1,\dots x_n)|
|f(x_{\pi(1)},\dots,x_{\pi(n)})|  \label{5.7} \\
&&\qquad\qquad \prod_{s=1}^n 
\left(g_{j_s,j_s}(x_s) g_{j_{\pi^{-1}(s)},j_{\pi^{-1}(s)}}(x_s)\right)^{1/2}
\mu(\,dx_1)\dots\mu(\,dx_n). \nonumber
\end{eqnarray}
On the other hand, we get with the help of the Schwarz inequality that
\begin{eqnarray}
&&\!\!\!\!\!\!\!\!\!\!\!\!\!\!
\int |f(x_1,\dots x_n)||f(x_{\pi(1)},\dots,x_{\pi(n)})|
\prod_{s=1}^n \left(g_{j_s,j_s}(x_s) g_{j_{\pi^{-1}(s)},j_{\pi^{-1}(s)}}(x_s)\right)^{1/2}
\nonumber  \\
&&\qquad\qquad\qquad\qquad\qquad\qquad \qquad\qquad\qquad
\mu(\,dx_1)\dots \mu(\,dx_n) 
\label{5.8} \\
&&\!\!\!\!\!\!\le \left(\int |f(x_1,\dots x_n)|^2
\prod_{s=1}^n g_{j_s,j_s}(x_s)\mu(\,dx_1)\dots \mu(\,dx_n)\right)^{1/2} 
\nonumber \\ 
&&\left(\int |f(x_{\pi(1)},\dots,x_{\pi(n)})|^2
\prod_{s=1}^n g_{j_{\pi^{-1}(s)},j_{\pi^{-1}(s)}}(x_s)
\mu(\,dx_1)\dots \mu(\,dx_n)\right)^{1/2}  
\nonumber 
\end{eqnarray}
for all $\pi\in\Pi(n)$. Let us also observe that the map $T$ 
from ${\mathbb R}^{n\nu}$ to ${\mathbb R}^{n\nu}$, defined as 
$$
T(x_1,\dots,x_n)=(x_{\pi(1)},\dots,x_{\pi(n)})
$$ 
is a bijection, and it is a measure preserving transformation from 
$$
({\mathbb R}^{n\nu}, G_{j_1,j_1}\times\cdots\times G_{j_n,j_n})=
({\mathbb R}^{n\nu}, g_{j_1,j_1}(x_1)\cdots g_{j_n,j_n}(x_n)\mu(\,dx_1)
\dots\mu(\,dx_n)\,)
$$
to 
\begin{eqnarray*}
&&({\mathbb R}^{n\nu}, G_{j_{\pi^{-1}(1)} ,j_{\pi^{-1}(1) } }\times\cdots\times 
G_{j_{\pi^{-1}(n)},j_{\pi^{-1}(n)}})\\
&&\qquad=({\mathbb R}^{n\nu}, g_{j_{\pi^{-1}(1)},j_{\pi^{-1}(1)}}(x_1)
\cdots g_{j_{\pi^{-1}(n)},j_{\pi^{-1}(n)}}(x_n)\mu(\,dx_1)\dots\mu(\,dx_n)\,).
\end{eqnarray*}
To see this it is enough to check that if $A=A_1\times\cdots\times A_n$,
then 
$$
(G_{1,1}\times \cdots\times G_{n,n})(A)=\prod_{l=1}^nG_{l,l}(A_l),
$$
$TA=A_{\pi^{-1}(1)}\times\cdots\times A_{\pi^{-1}(n)}$,
\begin{eqnarray*}
&&(G_{j_{\pi^{-1}(1)} ,j_{\pi^{-1}(1) } }\times\cdots\times 
G_{j_{\pi^{-1}(n)},j_{\pi^{-1}(n)}})(TA) \\
&&\qquad=\prod_{l=1}^n
G_{j_{\pi^{-1}(l)},j_{\pi^{-1}(l)}}(A_{\pi^{-1}(l)})=
(G_{1,1}\times \cdots\times G_{n,n})(A).
\end{eqnarray*}
The last identity together with the bijective property of $T$
imply that it is measure preserving.

Because of the measure preserving property of the operator~$T$ 
we can write that
\begin{eqnarray}
&&\int |f(x_1,\dots x_n)|^2 \prod_{s=1}^n g_{j_s,j_s}(x_s)
\mu( \,dx_1)\dots \mu(\,dx_n)  
\label{5.9} \\
&&\qquad=  \int |f(x_{\pi(1)},\dots,x_{\pi(n)})|^2
\prod_{s=1}^n g_{j_{\pi^{-1}(s)},j_{\pi^{-1}(s)}}(x_s)\mu( \,dx_1)\dots\mu( \,dx_n).
\nonumber 
\end{eqnarray}
Relation (\ref{5.6}) follows from relations 
(\ref{5.7}), (\ref{5.8}) and (\ref{5.9}).

To prove formulas (\ref{5.4}) and (\ref{5.5}) first we 
prove the following relations. Let a regular system
${\cal D}=\{\Delta_k,\;k=\pm1,\pm2,\dots,\pm N\}$ be given, choose an
integer $n\ge 1$, some numbers $j_1,\dots,j_n$ and $j_1'\dots,j'_n$ 
such that $1\le j_s,j'_s\le d$, $1\le s\le d$, together with two 
sequences of numbers $k_1,\dots,k_n$ and $l_1,\dots,l_n$ such that
$k_s,l_s\in\{\pm1,\dots,\pm N\}$ for all $1\le s\le n$, and they also
satisfy the relation $k_s\neq\pm k_{s'}$, and $l_s\neq\pm l_{s'}$ if 
$s\neq s'$. I claim that under these conditions
\begin{equation}
EZ_{G,{j_1}}(\Delta_{k_1})\cdots Z_{G,{j_n}}(\Delta_{k_n})
\overline{Z_{G,{j'_1}}(\Delta_{l_1})\cdots Z_{G,{j'_n}}(\Delta_{l_n})}=0 
\label{5.10}
\end{equation}
if $\{k_1,\dots,k_n\}\neq\{l_1,\dots,l_n\}$. On the other hand,
if 
\begin{equation}
l_p=k_{\pi(p)} \;\;\textrm{for all } 1\le p\le n \label{5.11} 
\end{equation}
with some permutation $\pi\in\Pi(n)$, then
\begin{eqnarray}
&&EZ_{G,{j_1}}(\Delta_{k_1})\cdots Z_{G,{j_n}}(\Delta_{k_n})
\overline{Z_{G,{j'_1}}(\Delta_{l_1})\cdots Z_{G,{j'_n}}(\Delta_{l_n})} 
\nonumber \\
&&\qquad= G_{j_1,j'_{\pi^{-1}(1)}}(\Delta_{k_1})\cdots G_{j_n,j'_{\pi^{-1}(n)}}
(\Delta_{k_n}).
\label{5.12}
\end{eqnarray}
Let me remark that there cannot be two different permutations
$\pi\in\Pi(n)$ satisfying relation~(\ref{5.11}), since by our
assumption also elements of the set $\{k_1,\dots,k_n\}$ are 
different, and the same relation holds for the set
$\{1_1,\dots,l_n\}$.

To prove (\ref{5.10}) we show that under its conditions the product
$$
Z_{G,{j_1}}(\Delta_{k_1})\cdots Z_{G,{j_n}}(\Delta_{k_n})
\overline{Z_{G,{j'_1}}(\Delta_{l_1})\cdots Z_{G,{j'_n}}(\Delta_{l_n})}
$$
can be written in the form of a product of two independent 
terms in such a way that one of them has expectation zero. 

Since $\{k_1,\dots,k_n\}\neq\{l_1,\dots,l_n\}$, there is 
such an element $k_s$ for which $k_s\neq l_t$ for all 
$1\le t\le n$, and also the relation $k_s\neq \pm k_t$ if 
$s\neq t$, holds. If the relation $k_s\neq \pm l_t$ also
holds for all $1\le t\le n$, then $Z_{G,{j_s}}(\Delta_{k_s})$ 
is independent of the product of the product of the 
remaining terms in this product because of property~(vi) 
of vector valued random spectral measures given in 
Section~3, and $EZ_{G,{j_s}}(\Delta_{k_s})=0$. Hence 
relation~(\ref{5.10}) holds in this case.

In the other case, there is an index $s'$ such that 
$l_{s'}=-k_s$. In this case the vector 
\begin{eqnarray*}
(Z_{G,{j_s}}(\Delta_{k_s}),\overline{Z_{G,{j_{s'}}}(\Delta_{l_{s'}})})
&=&(Z_{G,{j_s}}(\Delta_{k_s}),Z_{G,{j_{s'}}}(-\Delta_{l_{s'}})) \\
&=&(Z_{G,{j_s}}(\Delta_{k_s}),Z_{G,{j_{s'}}}(\Delta_{k_{s}}))
\end{eqnarray*}
is independent of the remaining terms, (because of 
property~(vi) of the vector valued random spectral measures). 
In last the relation we exploited that 
$-\Delta_{l_{s'}}=\Delta_{k_s}$). Hence
$$
EZ_{G,{j_s}}(\Delta_{k_s})\overline{Z_{G,{j_{s'}}}(\Delta_{l_{s'}})}=
EZ_{G,{j_s}}(\Delta_{k_s})\overline{Z_{G,{j_{s'}}}(-\Delta_{k_s})}=0,
$$
and relation (\ref{5.10}) holds in this case, too.

To prove (\ref{5.12}) let us observe that under its condition 
the investigated product can be written in the form
\begin{eqnarray*}
&&Z_{G,{j_1}}(\Delta_{k_1})\cdots Z_{G,{j_n}}(\Delta_{k_n})
\overline{Z_{G,{j'_1}}(\Delta_{l_1})\cdots Z_{G,{j'_n}}(\Delta_{l_n})} \\
&&\qquad =\prod_{p=1}^n Z_{G,{j_p}}(\Delta_{k_p}) 
\overline{Z_{G,{ j'_{\pi^{-1}(p)}}}(\Delta_{k_p})}.
\end{eqnarray*}
The terms in the product at the right-hand side are independent 
for different indices~$s$, and
$EZ_{G,{j_p}}(\Delta_{k_p})\overline{Z_{G,{ j'_{\pi^{-1}(p)}}}(\Delta_{k_p})}
=G_{j_p,j'_{\pi^{-1}(p}}(\Delta_{k_p})$. Formula~(\ref{5.12}) follows from 
these relations and the independence between the terms in the
last product. (Here we use again property~(vi) of the random
spectral measures.)

To prove formula (\ref{5.4}) let us take a regular system 
$$
{\cal D}=\{\Delta_k,\;k=\pm1,\dots,\pm N\}
$$ 
such that both functions $f$ and $h$ are adapted to it. This can 
be done by means of a possible refinement of the original 
regular systems corresponding to the functions $f$ and 
$h$. Then we can write by exploiting (\ref{5.2}) and (\ref{5.10}) that
\begin{eqnarray*}
&&EI_n(f|j_1,\dots,j_n)I_n(h(|j'_1,\dots,j'_n) 
=EI_n(f|j_1,\dots,j_n)\overline{I_n(h(|j'_1,\dots,j'_n)} \\
&&\qquad =\sum_{\pi\in\Pi(n)} 
\sum_{\substack{(k_1,\dots k_n),\; (l_1,\dots l_n)\\
k_p=\pm1,\dots,\pm N,\; p=1,\dots, n\\
l_p=k_{\pi(p)} \;p=1,\dots,n}}
f(u_{k_1},\dots u_{k_n})
\overline{h(u_{k_{\pi(1)}},\dots,u_{k_{\pi(n) }}) } \\
&&\qquad\qquad\qquad\qquad 
E Z_{G,{j_1}}(\Delta_{k_1})\cdots Z_{G,{j_n}}(\Delta_{k_n})
\overline{Z_{G,{j'_1}}(\Delta_{l_1})\cdots Z_{G,{j'_n}}(\Delta_{l_n})},
\end{eqnarray*}
where $u_k\in\Delta_k$ for all $k=\pm1,\dots,\pm N$.

The expected value of the product at the right-hand side of 
this identity can be calculated with the help of~(\ref{5.12}), 
and this yields that
\begin{eqnarray*}
&&EI_n(f|j_1,\dots,j_n)I_n(h(|j'_1,\dots,j'_n) \\
&&\qquad =\sum_{\pi\in\Pi(n)} 
\sum_{\substack{(k_1,\dots k_n),\; (l_1,\dots l_n)\\
k_p=\pm1,\dots,\pm N,\; p=1,\dots, n\\
l_p=k_{\pi(p)}, \;p=1,\dots,n}}
f(u_{k_1},\dots u_{k_n})
\overline{h(u_{l_1},\dots,u_{l_n}}) \\
&&\qquad\qquad\qquad\qquad 
G_{j_1,j'_{\pi^{-1}(1)}}(\Delta_{k_1})\cdots G_{j_n,j'_{\pi^{-1}(n)}}(\Delta_{k_n}) \\
&&\qquad =\sum_{\pi\in\Pi(n)} \int f(x_1,\dots x_n)
\overline{h(x_{\pi(1)},\dots,x_{\pi(n)})} \\
&&\qquad\qquad\qquad G_{j_1,j'_{\pi^{-1}(1)}}(\,dx_1)\dots G_{j_n,j'_{\pi^{-1}(n)}}(\,dx_n).
\end{eqnarray*}
Formula~(\ref{5.4}) is proved.

The proof of~(\ref{5.5}) is based on a similar idea, but it 
is considerably simpler. It can be proved similarly 
to relation~(\ref{5.10}) that for $n\neq n'$
\begin{equation}
EZ_{G,{j_1}}(\Delta_{k_1})\cdots Z_{G,{j_n}}(\Delta_{k_n})
\overline{Z_{G,{j'_1}}(\Delta_{l_1})\cdots 
Z_{G,{j'_{n'}}}(\Delta_{l_{n'}})}=0 \label{5.13}
\end{equation}
if we define this expression by means a regular system
$$
{\cal D}=\{\Delta_k,\;k=\pm1,\pm2,\dots,\pm N\},
$$ 
some numbers
$j_1,\dots,j_n$ and $j_1'\dots,j'_{n'}$, all of them between 
1 and $d$, together with two sequences of numbers 
$k_1,\dots,k_n$ and $l_1,\dots,l_{n'}$ such that
$k_s,l_s\in\{\pm1,\dots,\pm N\}$ for all these numbers, and they
satisfy the relation $k_s\neq\pm k_{s'}$, and $l_s\neq\pm l_{s'}$ if 
$s\neq s'$. Then, if we express
$$
EI_n(f|j_1,\dots,j_n)I_{n'}(h(|j'_1,\dots,j'_{n'}) 
=EI_n(f|j_1,\dots,j_n)\overline{I_{n'}(h(|j'_1,\dots,j'_{n'})} 
$$
similarly as we have done in the proof of~(\ref{5.12}) we get 
such a sum where all terms equal zero because of~(\ref{5.13}). 
This implies relation~(\ref{5.5}).

To define the Wiener--It\^o integral for all functions 
$f\in{\cal K}_{n,j_1,\dots,j_n}$ we need the following result.

\medskip\noindent
{\bf Lemma 5.1.} {\it The class of simple functions
$\hat{\cal K}_{n,j_1,\dots,j_n}$ is a dense linear subspace
of the (real) Hilbert space ${\cal K}_{n,j_1,\dots,j_n}$.}

\medskip
Lemma 5.1 is the multivariate version of Lemma~4.1 in~\cite{11}. 
(A more transparent proof of this result was given in the 
Appendix of~\cite{12}.) Actually, we do not have to prove Lemma~5.1, 
because it simply follows from Lemma~4.1 of~\cite{11}. By applying 
this result for $G=\sum_{j=1}^n G_{j,j}$ we get that all bounded 
functions of ${\cal K}_{n,j_1,\dots,j_n}$ are in the closure 
of $\hat{\cal K}_{n,j_1,\dots,j_n}$. But this implies that all 
functions of ${\cal K}_{n,j_1,\dots,j_n}$ are in this closure.

\medskip
Let us take the $L_2$ norm in the Hilbert space ${\cal H}$.
Then we have for all $f\in\hat{\cal K}_{n,j_1,\dots,j_n}$
$I_n(f|j_1,\dots,j_n)\in{\cal H}$, and by formula~(\ref{5.6}) 
$$
\|I_n(f|j_1,\dots,j_n)\|=\left[E(I_n(f|j_1,\dots,j_n)^2)\right]^{1/2}
\le \sqrt{n!}\|f_{n,j_1,\dots,j_n}\|.
$$
Hence Lemma~5.1 enables us to extend the Wiener--It\^o 
integral $I_n(f|j_1,\dots,j_n)$ for all
$f\in{\cal K}_{n,j_1,\dots,j_n}$. Moreover, relations 
(\ref{5.2})---(\ref{5.6}) remain valid in the Hilbert space 
${\cal K}_{n,j_1,\dots,j_n}$ after this extension.

\medskip\noindent
{\it Remark.} In~(\ref{5.6}) we have given an upper bound for 
the second moment of a multiple Wiener--It\^o integral, but we
cannot write equality in this formula. In the scalar-valued 
case we had an identity in the corresponding relation. At 
least this was the case if we took the Wiener--It\^o integral 
of a symmetric function. On the other hand, working only 
with Wiener--It\^o  integrals of symmetric functions did 
not mean a serious restriction. This relative weakness of 
formula~(\ref{5.6}) (the lack of identity) is the reason 
why we cannot represent such a large class of random 
variables in the form of a sum of Wiener--It\^o integrals 
as in the scalar valued case. (See the discussion in 
Section~7 about this problem.)

\medskip
I would mention that there is a slightly stronger version of
Lemma 5.1 which is useful in the study of the last section of
this paper, when we are interested in the question under what
conditions we can state that a sequence of Wiener--It\^o integrals
converges to a Wiener--It\^o integral. Here is this result.

\medskip\noindent
{\bf Lemma 5.2.} {\it For all functions 
$f\in{\cal K}_{n,j_1,\dots,j_n}$ and numbers $\varepsilon>0$ there 
is such a simple function $g\in\hat{\cal K}_{n,j_1,\dots,j_n}$ for
which $\|f-g\|\le\varepsilon$ in the norm of the Hilbert space 
${\cal K}_{n,j_1,\dots,j_n}$, and there is a regular system 
${\cal D}=\{\Delta_k,\;k=\pm1,\pm2,\dots,\pm N\}$ to which the
function $g$ is adapted, and the boundary of all sets
$\Delta_k\in{\cal D}$ has zero $\mu$-probability with the measure
$\mu$ we chose as the dominating measure for the complex measures
$G_{j,j'}$ in our considerations.}

\medskip
Lemma~5.2 also follows from the results of~\cite{11} or~\cite{12}.

\medskip
Finally, I would make the following small remark. If we define 
a new function by reindexing the variables of a function of 
$h\in{\cal K}_{n,j_1,\dots,j_n}$ by means of a permutation of the 
indices, and we change the indices of the spectral measure 
$Z_{G,{j_s}}$ in the Wiener-It\^o integral $I_n(h|j_1,\dots,j_n)$
in an appropriate way, then we get a new Wiener--It\^o 
integral whose value agrees with the original integral 
$I_n(h|j_1,\dots,j_n)$. More explicitly, the following result
holds.

\medskip\noindent
{\bf Lemma 5.3.} {\it Given a function $h\in{\cal K}_{n,j_1,\dots,j_n}$  
and a permutation $\pi\in\Pi(n)$ define the function
$h_\pi(x_1,\dots,x_n)=h(x_{\pi(1)},\dots,x_{\pi(n)})$. The following
identity holds. 
\begin{eqnarray}
&&\int h(x_1,\dots,x_n)Z_{G,{j_1}}(\,dx_1)\dots Z_{G,{j_n}}(\,dx_n) \nonumber \\
&&\qquad 
=\int h_\pi(x_1,\dots,x_n)
Z_{G,{j_{\pi(1) }}}(\,dx_1)\dots Z_{G,{j_{\pi(n)}}}(\,dx_n).
\label{5.14}
\end{eqnarray}
(In particular, $h_\pi\in{\cal K}_{n,j_{\pi(1)},\dots,j_{\pi(n)}}$, thus the 
integrals on both sides of the identity are meaningful.)}

\medskip\noindent
{\it Proof of Lemma 5.3.} This identity can be simply checked if 
$h$ is a simple function. It is enough to observe that if
$h(x_1,\dots,x_n)=h_1(x_1)\cdots h_n(x_n)$ with some 
$x_l\in\Delta_{k_l}$, $g(_l(\cdot)$ is some function on 
${\mathbb R}^{\nu}$, $1\le  l\le n$, then
$$
\int h(x_1,\dots,x_n)Z_{G,{j_1}}(\,dx_1)\dots Z_{G,{j_n}}(\,dx_n) 
=\prod_{l=1}^n h_l(x_l)Z_{G,j_l}(\Delta_{k_l}),
$$
$h_\pi(x_1,\dots,x_l)=h_1(x_{\pi_1})\cdots h_n(x_{\pi_n})$, 
$$
\int h_\pi(x_1,\dots,x_n)
Z_{G,{j_{\pi(1) }}}(\,dx_1)\dots Z_{G,{j_{\pi(n)}}}(\,dx_n)=
\prod_{l=1}^n h(x_{\pi_l})Z_{G,j_{\pi_l}}(\Delta_{k_{\pi(l)}}),
$$
and the last two Wiener--It\^o integrals equal. Then a 
simple limiting procedure implies it in the general case. 
Lemma~5.3 is proved.

\medskip
We saw in~\cite{11} that in the scalar valued case the value 
of a Wiener--It\^o integral 
$\int f(x_1,\dots,x_n)Z_G(\,dx_1)\dots Z_G(\,dx_n)$ 
does not change if we replace the kernel function~$f$ by 
the function we get by permuting its variables $x_1,\dots,x_n$ 
in an arbitrary way. Lemma~5.3 is the generalization of this 
result to the case when we integrate with respect to the 
coordinates of a vector-valued random spectral measure. 

\medskip\noindent
{\it Remark.} A consequence of the result of Lemma~5.3 shows
an essential difference between the behaviour of multiple
Wiener--It\^o integrals with respect to scalar and vector 
valued random spectral measures. It follows from the scalar
valued version of Lemma~5.3 that in the scalar valued case
the Wiener--It\^o integral of a kernel function agrees with
the Wiener--it\^o integral of the symmetrization of this
kernel function. This has the consequence that in the scalar
valued case we can restrict our attention to the Wiener--It\^o
integrals of symmetrical functions which do not change their
values by any permutation of their variables. It can be seen
that any random variable which can be written as the sum 
of Wiener--It\^o integrals can be written in a unique form
as a sum of Wiener--It\^o integrals of different multiplicity
with symmetric kernel functions. The analogous result does 
not hold in the vector valued case. Indeed, if there is 
some linear dependence among the the coordinates of the
underlying vectors in a vector valued stationary random
field, then such functions $f_j$ can be found for
which $\sum_{j=1}^d \int f_j(x)Z_{G,j}(\,dx)\equiv0$,
and not all kernel functions $f_j$ disappear in the above
sum. This shows that the unique representation of the
random variables by means of a sum of Wiener--It\^o 
integrals may not hold in vector valued models.

\section{The diagram formula for the product of multiple 
Wiener--It\^o integrals}

Let us consider a vector valued random spectral measure 
$(Z_{G,1,}\dots,Z_{G,d})$ corresponding to the matrix valued 
spectral measure $(G_{j,j'})$, $1\le j,j'\le d$, of a vector 
valued stationary Gaussian random field with expectation zero 
(either to a discrete random field 
$X(p)=(X_1(p),\dots,X_d(p))$, $p\in{\mathbb Z}^\nu$, or to a 
generalized one $X(\varphi)=(X_1(\varphi),\dots,X_d(\varphi))$, 
$\varphi\in{\cal S}^\nu$). Let us assume that the spectral 
measure $G_{j,j'}$, $1\le j,j'\le d$, is non-atomic, and take 
two Wiener--It\^o integrals 
\begin{equation}
I_n(h_1|j_1,\dots,j_n)=\int h_1(x_1,\dots,x_n)Z_{G,{j_1}}(\,dx_1)\dots 
Z_{G,{j_n}}(dx_n) \label{6.1}
\end{equation}
and
\begin{equation}
I_mh_2|j'_1,\dots,j'_m)=\int h_2(x_1,\dots,x_m)Z_{G,{j'_1}}(\,dx_1)\dots 
Z_{G,{j'_m}}(dx_m) \label{6.2}
\end{equation}
with some kernel functions $h_1\in{\cal K}_{n,j_1,\dots,j_n}$ and 
$h_2\in{\cal K}_{m,j'_1,\dots,j'_m}$, where $j_s, j'_t\in\{1,\dots,d\}$
for all $1\le s\le n$ and $1\le t\le m$.

Actually we formulate our problems in a slightly different 
form which is more appropriate for our discussion. We take two 
functions $h_1(x_1,\dots,x_n)$ and $h_2(x_{n+1},\dots,x_{n+m})$ 
in the space ${\mathbb R}^{(n+m)\nu}$, and define the function 
\hfill\break 
$h^{(0)}_2(x_1,\dots,x_m)$ by the identity
$$
h^{(0)}_2(x_1,\dots,x_m)=h_2(x'_{n+1},\dots,x'_{n+m}))
\textrm{ if } (x_1,\dots,x_m)=(x'_{n+1},\dots,x'_{n+m}).
$$
We assume that $h_1\in{\cal K}_{n,j_1,\dots,j_n}$,
$h^{(0)}_2\in{\cal K}_{m,j'_1,\dots,j'_m}$. Then we define the 
Wiener--It\^o integrals~(\ref{6.1}) and~(\ref{6.2}) with 
the kernel functions $h_1$ and $h_2^{(0)}$. In 
formula~(\ref{6.2}) we  should have written the function 
$h_2^{(0)}$, but we omitted the superscript~$^{(0)}$.

I shall present a result in which we express the product of these 
two Wiener--It\^o integrals as a sum of Wiener--It\^o integrals with
different multiplicities. This result is called the diagram formula, 
since the kernel functions of the Wiener--It\^o integrals appearing 
in this sum are expressed by means of some diagrams. This result is 
a multivariate version of the diagram formula proved in Chapter~5 
of \cite{11}. In that work also the product of more than two 
Wiener--It\^o integrals is expressed in the form of a sum of 
Wiener--integrals. But actually the main point of the proof is 
to show the validity of the diagram formula for the product of 
two Wiener--It\^o integrals, and we shall need only this result. 
So I restrict my attention only to this case. Actually we need 
the diagram formula only in a special case. The result in this 
special case will be given in a corollary.

To express the product of the two Wiener--It\^o integrals 
in formulas~(\ref{6.1}) and~(\ref{6.2}) as a sum of 
Wiener--It\^o integrals first I introduce a class of coloured 
diagrams $\Gamma=\Gamma(n,m)$ that will be used in the 
definition of the Wiener--It\^o integrals we shall be working 
with. A coloured diagram $\gamma\in\Gamma$ is a graph whose 
vertices are the pairs of integers $(1,s)$, $1\le s\le n$, and 
$(2,t)$, $1\le t\le m$. Each vertex is coloured with one of 
the numbers $1,\dots,d$. The colour of the vertex $(1,s)$ is 
$j_s$, $1\le s\le n$, and the colour of the vertex $(2,t)$ is 
$j'_t$, $1\le t\le m$. The set of vertices of the form $(1,s)$ 
will be called the first row and the set of vertices of the 
form $(2,t)$ will be called the second row  of a diagram 
$\gamma\in\Gamma$. The coloured diagrams $\gamma\in\Gamma$ are 
those undirected graphs with the above coloured vertices for 
which edges can go only between vertices of the first and 
second row, and from each vertex there starts zero or one edge. 
Given a coloured diagram $\gamma\in\Gamma$ we shall denote 
the number of its edges by $|\gamma|$.

I shall define for all coloured diagrams $\gamma\in\Gamma$ a
multiple Wiener--It\^o integral depending on~$\gamma$. The
diagram formula states that the product of the Wiener--It\^o 
integrals in~(\ref{6.1}) and~(\ref{6.2}) equals the sum of 
these Wiener--It\^o integrals. 

In the formulation of the diagram formula I shall work
with the functions $h_1(x_1,\dots,x_n)$ and 
$h_2(x_{n+1},\dots,x_{n+m})$ in ${\mathbb R}^{n+m}$.
The function \hfill\break 
$h_2(x_{n+1},\dots,x_{n+m})$ 
is the function which appeared in the definition of
the kernel function $h^{(0)}_2(x_1,\dots,x_m)$ in the
Wiener--It\^o integral in~(\ref{6.2}). We define
with their help the function
\begin{equation}
H(x_1,\dots,x_{n+m})=h_1(x_1,\dots,x_n)h_2(x_{n+1},\dots,x_{n+m}). 
\label{6.3}
\end{equation}
We shall define the kernel functions appearing in the 
Wiener--it\^o integrals in the diagram formula with the
help of the functions $H(x_1,\dots,x_{n+m})$. In the definition
of these kernel functions I shall apply the following 
natural bijection~$S$ between the coordinates of the vectors 
in~${\mathbb R}^{n+m}$, i.e. the set $\{1,\dots,n+m\}$ and
the vertices of the diagrams of $\gamma\in\Gamma$. 
\begin{equation}
S((1,k))=k \textrm{ for } 1\le k\le n, \quad\textrm{and} \quad
S((2,k))=n+k \textrm{ for } 1\le k\le m. \label{6.4}
\end{equation} 
To simplify the formulation of the diagram formula 
I shall introduce the following notation with the
help of the colours of the diagrams.
\begin{equation}
J(1,k)=j_k,\;\; 1\le k\le n \quad\textrm{and}\quad
J(2,l)=j'_l, \;\; 1\le l\le m. \label{6.5}
\end{equation}

First I give the formal definition of the Wiener--It\^o 
integrals that appear in the diagram formula, and then I 
give an informal explanation of this definition by briefly 
indicating the picture behind it. Then I describe the 
diagram formula with the help of the Wiener--It\^o integrals
corresponding to the diagrams $\gamma\in\Gamma$. To 
explain this result better I shall present an example 
after its formulation, where the product of two
Wiener--It\^o integrals is considered, and I show how to 
calculate a typical term in the sum of Wiener--It\^o 
integrals which appears if we apply the diagram formula
for this product.

Let us fix some diagram $\gamma\in\Gamma$. I explain how 
to define the the Wiener--It\^o integral corresponding to 
$\gamma$ in the diagram formula. First I define a function 
$H_\gamma(x_1,\dots,x_{n+m})$ which we get by means of an
appropriate permutation of the indices of the function~$H$
defined in (\ref{6.3}). This permutation of the indices
depends on the diagram~$\gamma$.

To define this permutation of the indices first
I define a map $T_\gamma$ which maps the set $\{1,\dots,n+m\}$ 
to the elements in the rows of the diagrams. This map depends 
on the diagram~$\gamma$. 

To define this map first I introduce the 
following sets depending on the diagram $\gamma$:
\begin{eqnarray}
A_1=A_1(\gamma)&=&\{r_1,\dots,r_{n-|\gamma|}\colon\; 1\le r_1<r_2<\cdots
<r_{n-|\gamma|}\le n   \label{6.6} \\ 
&&\qquad\qquad\textrm{no edge of } \gamma \textrm{ starts from }
(1, r_k),\quad 1\le k\le n-|\gamma|\}, \nonumber 
\end{eqnarray}
\begin{eqnarray}
A_2=A_2(\gamma)&=&\{t_1,\dots,t_{m-|\gamma|}\colon\; 1\le t_1<t_2<\cdots
<t_{m-|\gamma|}\le m  \label{6.7} \\ 
&&\qquad\qquad\textrm{no edge of } \gamma \textrm{  starts from }(2,t_k),
\quad 1\le k\le m-|\gamma|\}  \nonumber
\end{eqnarray}
and
\begin{eqnarray}
B=B(\gamma)&=&\{(v_1,w_1),\dots, (v_{|\gamma|},w_{|\gamma|}))\colon\; 
1\le v_1<v_2<\cdots v_{|\gamma|}\le n  \nonumber \\
&&\qquad ((1,v_k),(2,w_k)) \textrm{ is an edge of } |\gamma|, 
\;\; 1\le k\le|\gamma|\}.
\label{6.8}
\end{eqnarray}
Let us also define with the help of the set $B$ the sets
\begin{equation}
B_1=B_1(\gamma)=\{v_1,\dots,v_{|\gamma|}\}, \quad B_2=B_2(\gamma)
=\{w_1,\dots,w_{|\gamma|}\} \label{6.9}
\end{equation}
with the numbers $v_k$ and $w_l$ appearing in the set
$$
B=B(\gamma)=\{(v_1,w_1)),\dots, (v_{|\gamma|},w_{|\gamma|}))\}.
$$
Now, I define the map  $T_\gamma$ in the following way.
\begin{eqnarray}
&&T_\gamma(k)=(1,r_k) \;\;\textrm{for} \;\;1\le k\le n-|\gamma|, 
\label{6.10} \\
&&T_\gamma(n-|\gamma|+k)=(2,t_k) \;\;\textrm{for} \;\;
1\le k\le m-|\gamma|, \nonumber  \\ 
&&T_\gamma(n+m-2|\gamma|+k)=(1,v_k) \;\;\textrm{for}\;\; 
1\le k \le |\gamma|,  \nonumber \\ 
&&T_\gamma(n+m-|\gamma|+k)=(2,w_k) \;\;\textrm{for}\;\; 
1\le k \le|\gamma|. \nonumber
\end{eqnarray}
In formula~(\ref{6.10}) we worked with the numbers $r_k$, $t_k$,
$v_k$ and $w_k$ defined in (\ref{6.6})---(\ref{6.9}). It 
has the following meaning. The first  $n-|\gamma|$ indices
are given in increasing order to the vertices from the first 
row from which no edge starts. Then the vertices of the
second row from which no edge starts get the next
$m-|\gamma|$ indices also in increasing order. Then the 
$|\gamma|$ vertices from the first row from which an edge 
starts get the subsequent $|\gamma|$ indices in increasing
order. The remaining $|\gamma|$ vertices from the second row 
from which an edge starts get the indices between 
$n+m-|\gamma|+1$ and $|n+m|$. They are indexed in such a way 
that if two vertices $(1,v_k)$ and $(2,w_k)$ are connected 
by en edge then the index of $(2,w_k)$ is obtained if we add 
$|\gamma|$ to the index of $(1,v_k)$. 

I define with the help of the function $T_\gamma$ and the
map  $S(\cdot)$ defined in (\ref{6.4}) the permutation
$$
\pi_\gamma(k)=S(T_\gamma(k)), \quad 1\le k\le n+m
$$
of the set $\{1,\dots,n+m\}$. Next I introduce the Euclidean
space ${\mathbb R}^{n+m}_\gamma$ with elements 
$x(\gamma)=(x(\gamma)_1,\dots,x(\gamma)_{n+m})$ by reindexing 
the arguments of the  Euclidean space  ${\mathbb R}^{n+m}$, 
where the functions $h_1(x_1,\dots,x_n)$ and 
$h_2(x_{n+1},\dots,x_{n+m})$ are defined in the following way.
$$
(x(\gamma)_1,\dots,x(\gamma)_{n+m})
=(x_{\pi_\gamma(1)},\dots,x_{\pi_\gamma(n+m)}) 
$$
with
$(x(\gamma)_1,\dots,x(\gamma)_{n+m})\in{\mathbb R}^{n+m}_\gamma$ 
and $(x_1,\dots,x_{n+m})\in{\mathbb R}^{n+m}$. It will be simpler 
to define the quantities needed in the definition of the 
Wiener--It\^o integral corresponding to the diagram~$\gamma$ 
as functions defined in the space $R^{n+n}_\gamma$. First we 
define the function $H_\gamma$ as
\begin{eqnarray}
&&
\!\!\!\!\!\!\!\!\!\!\!
H_\gamma(x(\gamma)_1,\dots,x(\gamma)_{n+m}) \label{6.11} \\ 
&&
\!\!\!\!\!\!\!\!\!\!\!
\qquad=H(x(\gamma)_1,\dots,x(\gamma)_{n-|\gamma|},
x(\gamma)_{n+m-2|\gamma|+1},\dots,x(\gamma)_{n+m-|\gamma|}, 
\nonumber\\
&&
\!\!\!\!\!\!\!\!\!\!\!
\qquad\qquad\qquad x(\gamma)_{n-|\gamma|+1},\dots,x(\gamma)_{n+m-2|\gamma|+1},
x(\gamma)_{(n+m-|\gamma|+1},\dots, x(\gamma)_{n+m}) \nonumber \\
&&
\!\!\!\!\!\!\!\!\!\!\!
\qquad=h_1(x(\gamma)_1,\dots,x(\gamma)_{n-|\gamma|},
x(\gamma)_{\pi_\gamma(n+m-2|\gamma|+1)},\dots,
x(\gamma)_{n+m-|\gamma|}) \nonumber \\
&&
\!\!\!\!\!\!\!\!\!\!\!
\qquad\qquad h_2(x(\gamma)_{n-|\gamma|+1},\dots,x(\gamma)_{n+m-2|\gamma|+1},
x(\gamma)_{n+m-|\gamma|+1},\dots, x(\gamma)_{n+m}).
\nonumber 
\end{eqnarray}

Next I define the function 
$\bar h_\gamma(x(\gamma)_1,\dots,x(\gamma)_{n+m-|\gamma|}))$ which 
we get by replacing the arguments $x(\gamma)_{n+m-|\gamma|+k}$ by 
$-x(\gamma)_{n+m-2|\gamma|+k})$ in the function $H_\gamma$ defined 
in formula~(\ref{6.11}) for all $1\le k\le \gamma$, i.e. I define
\begin{eqnarray}
&&
\!\!\!\!\!\!\!\!\!\!\!
\bar h_\gamma(x(\gamma)_1,\dots,x(\gamma)_{n+m-|\gamma|})  \label{6.12} \\
&&
\!\!\!\!\!\!\!\!\!\!\!
\qquad=H_\gamma(x(\gamma)_1,\dots,x(\gamma)_{n+m-|\gamma|},
-x(\gamma)_{n+m-2|\gamma|+1},\dots,-x(\gamma)_{n+m-|\gamma|})  \nonumber \\ 
&&
\!\!\!\!\!\!\!\!\!\!\!
\qquad=H(x(\gamma)_1,\dots,x(\gamma)_{n-|\gamma|},
x(\gamma)_{n+m-2|\gamma|+1},\dots,x(\gamma)_{n+m-|\gamma|}, 
\nonumber\\
&&
\!\!\!\!\!\!\!\!\!\!\!
\qquad\qquad\qquad x(\gamma)_{n-|\gamma|+1},\dots,x(\gamma)_{n+m-2|\gamma|+1},
\nonumber \\
&&
\!\!\!\!\!\!\!\!\!\!\!
\qquad\qquad\qquad\qquad 
-x(\gamma)_{n+m-2|\gamma|+1},\dots, -x(\gamma)_{n+m-|\gamma|}) 
\nonumber \\
&&
\!\!\!\!\!\!\!\!\!\!\!
\qquad=h_1(x(\gamma)_1,\dots,x(\gamma)_{n-|\gamma|},
x(\gamma)_{n+m-2|\gamma|+1},\dots,
x(\gamma)_{n+m-|\gamma|}) \nonumber \\
&&
\!\!\!\!\!\!\!\!\!\!\!
\qquad\qquad h_2(x(\gamma)_{n-|\gamma|+1},\dots,x(\gamma)_{n+m-2|\gamma|+1},
\nonumber \\
&&
\!\!\!\!\!\!\!\!\!\!\!
\qquad\qquad\qquad\qquad
-x(\gamma)_{n+m-2|\gamma|+1},\dots, -x(\gamma)_{n+m)-|\gamma|}).
\nonumber 
\end{eqnarray}
 In the next step I define the function
$\bar{\bar h}_\gamma(x(\gamma)_1,\dots,x(\gamma)_{n+m-2|\gamma|})$. 
This will be the kernel
function of the Wiener--It\^o integral which corresponds to
the diagram~$\gamma$ in the diagram formula if we
express it as a Wiener--It\^o integral with respect to the
variables $x(\gamma)_1,\dots,x(\gamma)_{n+m-2|\gamma|}$. 
\begin{eqnarray}
&&\bar{\bar h}_\gamma(x_\gamma)_1,\dots,x(\gamma)_{n+m-2|\gamma|}) 
=\int \bar h_\gamma(x(\gamma)_1,\dots,x(\gamma)_{n+m-|\gamma|}) \label{6.13} \\
&&\qquad\qquad \prod_{k=1}^{|\gamma|} 
G_{J(S^{-1}(n+m-2|\gamma|+k)),J(S^{-1}(n+m-|\gamma|+k))}
(\,dx(\gamma)_{n+m-2|\gamma|+k}) \nonumber \\
&&\qquad=\int \bar h_\gamma(x(\gamma)_1,\dots,x(\gamma)_{n+m-|\gamma|}) 
\prod_{k=1}^{|\gamma|} G_{j_{v_k},j'_{w_k}}(\,dx(\gamma)_{n+m-2|\gamma|+k}) \nonumber 
\end{eqnarray}
with the function $J(\cdot)$ defined in~(\ref{6.5}), the indices
$v_k$ and $w_k$ defined in~(\ref{6.8}) and the
function $T_\gamma$ defined in~(\ref{6.10}). 

I shall show that the Wiener--It\^o integrals
\begin{eqnarray}
&&I_{n+m-2|\gamma|}(\bar{\bar h}_\gamma|j_{r_1},\dots,j_{r_{n-|\gamma|}},
j'_{t_1},\dots,j'_{t_{m-|\gamma|}}) \label{6.14} \\
&&\qquad=\int \bar{\bar h}_\gamma(x(\gamma)_1,\dots,x(\gamma)_{n+m-2|\gamma|})  
\prod_{k=1}^{n+m-2|\gamma|}Z_{G,{J(S^{-1}(k))}}(dx(\gamma)_k)  \nonumber \\
&&\qquad=\int\bar{\bar h}_\gamma(x(\gamma)_1,\dots,x(\gamma)_{n+m-2|\gamma|}) 
\nonumber\\ 
&& \qquad\qquad\qquad \prod_{k=1}^{n-|\gamma|}Z_{G,{j_{r_k}}}(\,dx(\gamma)_k)
\prod_{l=1}^{m-|\gamma|}Z_{G,{j'_{t_l}}}(\,dx(\gamma)_{l+n-|\gamma|})
\nonumber 
\end{eqnarray}
exist for all $\gamma\in\Gamma$, and these Wiener--It\^o integrals
appear in the diagram formula. The numbers $r_k$ and $t_l$ in
this formula were defined in~(\ref{6.6}) and~(\ref{6.7}).

In formula~(\ref{6.14}) we integrated with respect to the coordinates
$x(\gamma)_s$, $1\le s\le n+m$, of the vectors in the Euclidean space 
${\mathbb R}^{n+m}_\gamma$. If we replace the variables $x(\gamma)_s$
by $x_s$ in~(\ref{6.14}), then we get a Wiener--it\^o integral in
the space ${\mathbb R}^{n+m}$  which has the same value. This means
that the following relation holds.
\begin{eqnarray}
&&I_{n+m-2|\gamma|}(\bar{\bar h}_\gamma|j_{r_1},\dots,j_{r_{n-|\gamma|}},
j'_{t_1},\dots,j'_{t_{m-|\gamma|}}) \label{6.15} \\
&&\qquad=I_{n+m-2|\gamma|}(h_\gamma|j_{r_1},\dots,j_{r_{n-|\gamma|}},
j'_{t_1},\dots,j'_{t_{m-|\gamma|}}) \nonumber \\
&&\qquad=\int h_\gamma(x_1,\dots,x_{n+m-2|\gamma|}) 
\nonumber\\ 
&& \qquad\qquad\qquad \prod_{k=1}^{n-|\gamma|}Z_{G,{j_{r_k}}}(\,dx_k)
\prod_{l=1}^{m-|\gamma|}Z_{G,{j'_{t_l}}}(\,dx_{l+n-|\gamma|})
\nonumber 
\end{eqnarray}
with
\begin{eqnarray*}
h_\gamma(x_1,\dots,x_{n+m-2|\gamma|})
&=&\bar{\bar h}_\gamma(x(\gamma)_1,\dots,x(\gamma)_{n+m-2|\gamma|}) \\
&=&\bar{\bar h}_\gamma(x_{\pi_\gamma(1)},\dots,x_{\pi_\gamma(n+m-2|\gamma|)}).
\end{eqnarray*}

Before describing the diagram formula I explain the content
of the above defined formulas.

Let us fix a diagram $\gamma\in\Gamma$, and let us call a 
vertex of it  from which no edge starts open, and a vertex 
from which an edge starts closed. We listed the open vertices 
from the first row in increasing order as 
$(1,r_1),\dots,(1,r_{n-|\gamma|})$, and the open vertices from 
the second row as $(2,t_1),\dots,(2,t_{m-|\gamma|})$. We 
listed the closed vertices from the first row in increasing 
order as $(1,v_1),\dots,(1,v_\gamma)$. Finally we listed the 
closed vertices from the second row as 
$(2,w_1),\dots,(2,w_\gamma)$, and we indexed them in such a way
that the vertices $(1,v_k)$ and $(2,w_k)$ are connected by
an edge for all $1\le k\le\gamma$.  

In formula~(\ref{6.10}) we defined the map $T_\gamma$ from 
the set $\{1,\dots,n+m\}$ to the set of vertices of the 
diagram~$\gamma$ with the help of the above listing of 
the vertices. First we considered the open vertices from 
the first row, then the open vertices from the second row, 
and then we finished with the closed vertices first from 
the first and then from the second row. We defined 
in~(\ref{6.11}) the permutation $\pi_\gamma$ of the set 
$\{1,\dots,n+m\}$ by applying first the map 
the map $T_\gamma$ and then the map $S$ defined~(\ref{6.4}).
We defined the function $H_\gamma$ in (\ref{6.12}) with 
the help of this permutation. We have introduced a 
Euclidean space ${\mathbb R}^{n+m}_\gamma$ whose elements 
we get by rearranging the indices of the coordinates of 
the Euclidean space ${\mathbb R}^{n+m}$ where we are 
working with the help of the permutation~$\pi_\gamma$,
and we have defined our functions in this space.

We defined the function $H_\gamma$ on the space 
${\mathbb R}^{n+m}_\gamma$ as the product of the functions $h_1$ 
and $h_2$ with reindexed variables. In the function $h_1$ 
first we took the variables $x(\gamma)_s=x_{\pi_\gamma(s)}$
with those indices $\pi_\gamma(s)$ which correspond to the
open vertices of the first row, and then the variables with 
indices corresponding to the closed vertices of the first row.
We defined the reindexation of the variables in the second
row similarly. First we took those variables whose indices
correspond to the open vertices and then the variables whose
indices correspond to the closed vertices of the second row. 

The variables 
$$
x(\gamma)_{n+m-2|\gamma|+k}=x_{\pi_\gamma(n+m-2|\gamma|+k)} \textrm{ and }
x(\gamma)_{n+m-|\gamma|+k}=x_{\pi_\gamma(n+m-|\gamma|+k)}
$$ 
in the function $H_\gamma$ are variables with indices 
corresponding to vertices connected by an edge. So in 
the definition of the function
$\bar h_\gamma$ in~(\ref{6.13}) I replaced in $H_\gamma$ the 
variable corresponding to the endpoint of an edge from 
the second row of the diagram~$\gamma$ by the variable 
corresponding to the other endpoint of this edge, and 
multiplied this variable by~$-1$. Thus the variables
$x(\gamma)_{n+m-2|\gamma|+k}=x_{\pi_\gamma(n+m-2|\gamma|+k)}$, 
$1\le k\le|\gamma$, of the function $\bar h_\gamma$ 
correspond to the edges of the diagram~$\gamma$. I 
defined the function $\bar{\bar h}_\gamma$ by integrating the 
function $\bar h_\gamma$ by these variables. The variable 
$x(\gamma)_{n+m-2|\gamma|+k}=x_{\pi_\gamma(n+m-2|\gamma|+k)}$ corresponds
to the $k$-th edge of the diagram, and we integrate this
variable with respect to the measure $G_{j_{v_k},j'_{w_k}}$,
that is with respect to the measure $G_{u,v}$ whose
coordinates are the colours of the endpoints of the
$k$-th edge. 

Finally we define the Wiener--It\^o integral corresponding
to the diagram $\gamma$ with kernel 
function~$\bar{\bar h}_\gamma$. We integrate the argument 
$x(\gamma)_k$ with respect to that random
spectral measure~$Z_{G,j}$ whose parameter agrees with
the colour of the vertex corresponding to this variable.
Thus we choose $Z_{G,j_{r_k}}(\,dx(\gamma)_k)$ for 
$1\le k\le n-|\gamma|$ and 
$Z_{G_{j'_{t_{k-n+|\gamma}}}}(\,dx(\gamma)_k)$ if 
$n-|\gamma|+1\le k\le n+m-2|\gamma|$. We can replace
this Wiener--It\^o integral defined in~(\ref{6.14}) 
with kernel function~$\bar{\bar h}_\gamma$ by the 
Wiener--It\^o integral defined in~(\ref{6.15}) with
kernel function~$h_\gamma$.

Next I formulate the diagram formula.

\medskip\noindent
{\bf Theorem 6.1. The diagram formula.} {\it Let us consider 
the Wiener--It\^o integrals $I_n(h_1|j_1,\dots,j_n)$ and 
$I_m(h_2|j_1',\dots,j_m')$ introduced in formulas (\ref{6.1}) and 
(\ref{6.2}). The following results hold.

\medskip
\begin{description}
\item[{\rm(A)}] The function $\bar{\bar h}_\gamma$ defined 
in~(\ref{6.13}) satisfies the relations
$$
\bar{\bar h}_\gamma\in {\cal K}_{n+m-2|\gamma|,
j_{r_1},\dots,j_{r_{n-|\gamma|}},j'_{t_1},\dots,j'_{t_{m-|\gamma|}}},
$$ 
and $\|\bar{\bar h}_\gamma\|\le \|h_1\| \|h_2\|$ for all $\gamma\in\Gamma$. 
Here the norm of  the function $h_1$ in ${\cal K}_{n,j_1,\dots,j_n}$, 
the norm of $\bar{\bar h}_2$ in ${\cal K}_{m,j'_1,\dots,j'_m}$, and 
the norm of $\bar{\bar h}_\gamma$ in 
${\cal K}_{n+m-2|\gamma|,
j_{r_1},\dots,j_{r_{n-|\gamma|}},j'_{t_1},\dots,j'_{t_{m-|\gamma|}}}$ is taken.

\item[{\rm(B)}]
\begin{eqnarray}
&&I_n(h_1|j_1,\dots,j_n)I_m(h_2|j'_1,\dots,j'_m) \label{6.16} \\
&&\qquad =\sum_{\gamma\in\Gamma}  
I_{n+m-2|\gamma|}(\bar{\bar h}_\gamma|j_{r_1},\dots,j_{r_{n-|\gamma|}},j'_{t_1},
\dots,j'_{t_{m-|\gamma|})}. \nonumber
\end{eqnarray}
The terms in the sum at the right-hand side of formula~(\ref{6.16}) 
were defined in formulas~(\ref{6.11})---(\ref{6.14}). The 
Wiener--It\^o integral
$$
I_{n+m-2|\gamma|}(\bar{\bar h}_\gamma|j_{r_1},\dots,j_{r_{n-|\gamma|}},j'_{t_1},
\dots,j'_{t_{m-|\gamma|})}
$$ 
in formula~(\ref{6.16}) can be replaced by the Wiener--It\^o integral
$$I_{n+m-2|\gamma|}(h_\gamma|j_{r_1},\dots,j_{r_{n-|\gamma|}},j'_{t_1},
\dots,j'_{t_{m-|\gamma|})}
$$ 
defined in~(\ref{6.15}).
\end{description}
}

\medskip
To understand the formulation of the diagram formula better 
let us consider the following example. We take a five 
dimensional stationary Gaussian random field with some 
spectral measure $G_{j,j'}(x)$, $1\le j,j'\le 5$, and random 
spectral measure $Z_{G,j}(\,dx)$, $1\le j\le 5$, corresponding 
to it. Let us understand how we define the Wiener--It\^o 
integral corresponding to a typical diagram when we apply 
the diagram formula in the following example. Take the 
product of two Wiener--It\^o integrals of the following form:
$$
I_3(h_1|2,3,5)=\int h_1(x_1,x_2,x_3)Z_{G,2}(\,dx_1)Z_{G,3}(\,dx_2)Z_{G,5}(\,dx_3)
$$
and
\begin{eqnarray*}
I_4(h_2|1,5,4,1)&=&\int h_2(x_1,x_2,x_3,x_4) \\
&& \qquad Z_{G.1}(\,dx_1)Z_{G,5}(\,dx_2) Z_{G,4}(\,dx_3)Z_{G,2}(\,dx_4),
\end{eqnarray*}
and let us write it in the form of a sum of Wiener--It\^o 
integrals with the help of the diagram formula.

First I give the vertices of the coloured diagrams we shall be 
working with together with their colours.

\medskip

\begin{figure}[h]
\begin{center}
\epsfig{file=arcon1.eps, width=5cm}
\end{center}
\caption{the vertices of the diagrams together with their colours}
%\label{arcon1.fig}
\end{figure}

\medskip

Next I consider a diagram~$\gamma$ which yields one of the terms 
in the sum  expressing the product of these two Wiener--It\^o 
integrals. I take the diagram which has two edges, one edge 
connecting the vertices $(1,2)$ and $(2,4)$, and another edge 
connecting the vertices $(1,3)$ and $(2,1)$. Let us calculate 
which Wiener--It\^o integral corresponds to this diagram~$\gamma$.

\medskip

\begin{figure}[h]
\begin{center}
\epsfig{file=arcon2.eps, width=5cm}
\end{center}
\caption{a typical diagram}
%\label{arcon1.fig}
\end{figure}

\medskip

Next I take this diagram~$\gamma$, and I show not only the 
indices and colours of its vertices, but for each vertex I 
also tell which value $T_\gamma(k)$ it equals. Here
$T_\gamma(k)$ is the function defined in formula~(\ref{6.10}). 

\medskip

\psfrag{(1,1),2}{$(1,1)=T_\gamma(1),2$}
\psfrag{(1,2),3}{$\;\; (1,2)=T_\gamma(4),3$}
\psfrag{(1,3),5}{$\quad \;\;(1,3)=T_\gamma(5),5$}
\psfrag{(2,1),1}{$(2,1)=T_\gamma(7),1$}
\psfrag{(2,2),5}{$(2,5)=T_\gamma(2),5$}
\psfrag{(2,3),4}{$(2,3)=T_\gamma(3),4$}
\psfrag{(2,4),2}{$(2,4)=T_\gamma(6),2$}

\begin{figure}[h]
\epsfig{file=arcon3.eps, width=10cm}
\caption{the previous diagram and the enumeration of their vertices with
the help of the function $T_\gamma$}
%\label{arcon1.fig}
\end{figure}

\medskip

To define the the Wiener--It\^o integral corresponding to this
diagram let us first consider the function
$$
H(x_1,\dots,x_7)=h_1(x_1,x_2,x_3)h_2(x_4,x_5,x_6,x_7)
$$
defined  in~(\ref{6.3}). Simple calculation shows that
the function $\pi_\gamma(\cdot)= S(T_\gamma(\cdot))$ has the 
following form in this example. $\pi_\gamma(1)=1$, 
$\pi_\gamma(2)=5$, $\pi_\gamma(3)=6$, $\pi_\gamma(4)=2$, 
$\pi_\gamma(5)=3$, $\pi_\gamma(6)=7$, $\pi_\gamma(7)=4$.
This also means that the coordinates of the vectors 
in the Euclidean space ${\mathbb R}^7_\gamma$ which we get 
by reindexing the coordinates of the vectors in
${\mathbb R}^7$ have the form
$$
(x(\gamma)_1,x(\gamma)_2,x(\gamma)_3,x(\gamma)_4,x(\gamma)_5,
x(\gamma)_6,x(\gamma)_7)=(x_1,x_5,x_6,x_2,x_3,x_7,x_4).
$$
Then we can write the function $\bar H_\gamma$ and $\bar h_\gamma$ 
defined in~(\ref{6.11}) and~(\ref{6.12}) as
$$
H_\gamma(x(\gamma)_1,\dots,x(\gamma)_7)
=h_1(x(\gamma)_1,x(\gamma)_4,x(\gamma)_5)
h_2(x(\gamma)_2,x(\gamma)_3,x(\gamma)_6,x(\gamma)_7),
$$
and
$$
\bar h_\gamma(x(\gamma)_1,\dots,x(\gamma)_5)
=h_1(x(\gamma)_1,x(\gamma)_4,x(\gamma)_5)
h_2(x(\gamma)_2,x(\gamma)_3,-x(\gamma)_4,-x(\gamma)_5).
$$
Then we have
$$
\bar{\bar h}_\gamma(x(\gamma)_1,x(\gamma)_2,x(\gamma)_3)
=\int\bar h_\gamma(x(\gamma)_1,\dots,x(\gamma)_5)G_{3,2}(\,dx(\gamma)_4)
G_{5,1}(\,dx(\gamma)_5),
$$
and 
\begin{eqnarray*}
&&I_3(\bar{\bar h}_\gamma|2,5,4) \\
&&\qquad
=\int \bar{\bar h}_\gamma(x(\gamma)_1,x(\gamma)_2,x(\gamma)_3)
Z_{G,2}(\,dx(\gamma)_1)Z_{G,5}(\,dx(\gamma)_2)Z_{G,4}(\,dx(\gamma)_3)
\end{eqnarray*}
is the multiple Wiener--It\^o integral corresponding to
the diagram~$\gamma$ in the diagram formula. To understand
the definition of the function $\bar{\bar h}_\gamma$ and of the
Wiener--It\^o integral $I_3(\bar{\bar h}_\gamma)$ let us
observe that the first edge of the diagram connects the
vertices $(1,2)$ and $(2,4)$ with colours 3 and 2, hence
in the definition of $\bar{\bar h}_\gamma$ we integrate the 
argument $x(\gamma)_4$ by $G_{3,2}(\,dx(\gamma)_4)$, the 
second edge connects the vertices $(1,3)$ and $(2,1)$ 
with colours 5 and 1, hence we integrate the variable
$x(\gamma)_5$ by $G_{5,1}(\,dx(\gamma)_5)$. In the definition
of the Wiener integral the variable $x(\gamma)_1$ corresponds
to the vertex $S^{-1}(\pi_\gamma(1))=(1,1)$ which has colour 2,
hence we integrate the variable $x(\gamma)_1)$ by 
$Z_{G,2}(\,dx(\gamma)_1)$. Similarly, we define the variable
$x(\gamma)_2$ by the measure determined by the colour of 
$S^{-1}(\pi_\gamma(2))=(2,2)$ which is 5, i.e. we integrate 
by $Z_{G,5}(\,dx(\gamma)_2)$. Finally $S^{-1}(\pi_\gamma(3))=(2,3)$ 
has colour 4, and we integrate the variable $x(\gamma)_3$ 
by $Z_{G,4}(\,dx(\gamma)_3)$.

The Wiener--It\^o integral
$I_3(\bar{\bar h}_\gamma|3,1,3)$ can be rewritten with the
help of formula~(\ref{6.15}) in the following form.
$$
I_3(\bar{\bar h}_\gamma|2,5,4)
=I_3(h_\gamma|2,5,4)
=\int h_\gamma(x_1,x_2,x_3)
Z_{G,2}(\,dx_1)Z_{G,5}(\,dx_2)Z_{G,4}(\,dx_3)
$$
with
$$
h_\gamma(x_1,x_2,x_3)=\int h_1(x_1,x_4,x_5)h_2(x_2,x_3,-x_4,-x_5)
G_{3,2}(\,dx_4) G_{5,1}(\,dx_5).
$$
This expression can be calculated similarly to
$I_3(\bar{\bar h}_\gamma|2,5,4)$, only we have to replace
$x(\gamma)_s$ everywhere by $x_s$ in the calculation.

\medskip\medskip
I formulate a Corollary of the diagram formula in which 
I consider that special case of this result when the second 
Wiener--It\^o integral defined in formula~(\ref{6.2}) is a 
one-fold integral. In this case it has the simpler form
\begin{equation}
I_1(h_2|j'_1)=\int h_2(x_1)Z_{G,{j'_1}}(\,dx_1) 
\quad\textrm{with }h_2\in{\cal K}_{1,j'_1}.  \label{6.17}
\end{equation}
Here again we formulate the problem in the following way. We 
take a pair of functions $h_1(x_1,\dots,x_n)$ and $h_2(x_{n+1})$ 
on ${\mathbb R}^{(n+1)\nu}$. Then we define a function 
$h_2^{(0)}(x_1)$ on ${\mathbb R}^1$ by the formula 
$h^{(0)}_2(x_1)=h_2(x_{n+1})$ if $x_1=x_{n+1}$. We integrate
the function $h_2^{(0)}(x)$ in formula~(\ref{6.17}), but we 
omit the superscript~${}^{(0)}$ in our notation. We assume 
that $h_1\in{\cal K}_{n,j_1,\dots,j_n}$, and $h_2\in{\cal K}_{1,j'_1}$. 

In the next Corollary I express the product of the 
Wiener--It\^o integrals given in~(\ref{6.1}) and (\ref{6.17}) 
as a sum of Wiener--It\^o integrals. This formula will be 
needed in the proof of the multivariate version of It\^o's 
formula in the next section. 

The diagram formula in this case has a simpler form, since the 
second row of the diagrams we are working with consists only of one 
point $(2,1)$. Hence there are only the diagram $\gamma_0\in\Gamma$ 
that contains no edges, and the diagrams $\gamma_p\in\Gamma$, 
$1\le p\le n$, which contain one edge that connects the vertices 
$(1,p)$ and $(2,1)$. 

\medskip\noindent
{\bf Corollary of Theorem 6.1.} {\it The product of the Wiener--It\^o 
integrals 
$$
I_n(h_1|j_1,\dots,j_n) \textrm{ \ and \ } I_1(h_2|j_1')
$$ 
introduced in formulas (\ref{6.1}) and (\ref{6.17}) satisfy the identity
\begin{eqnarray}
&&I_n(h_1|j_1,\dots,j_n)I_1(h_2|j_1') \label{6.18} \\
&&\qquad =\int h_{\gamma_0}(x_1,\dots,x_{n+1}) 
Z_{G,{j_1}}(\,dx_1)\cdots  Z_{G,{j_n}}(\,dx_n) 
Z_{G,{j'_1}}(\,dx_{n+1}) \nonumber \\
&&\qquad\qquad +\sum_{p=1}^n \int h_{\gamma_p}(x_1,\dots,x_{n-1})
\prod_{s=1}^{p-1}Z_{G,{j_s}}(\,dx_s) \prod_{s=p}^{n-1} 
Z_{G,{j_{s+1}}}(\,dx_s) \nonumber \\
&&\qquad=I_{n+1}(h_{\gamma_0}|j_1,\dots,j_n, j_1')
+\sum_{p=1}^n 
I_{n-1}(h_{\gamma_p}|j_1,\dots,j_{p-1},j_{p+1},\dots,j_n), \nonumber 
\end{eqnarray}
where $h_{\gamma_0}(x_1,\dots,x_{n+1})=h_1(x_1,\dots,x_n)h_2(x_{n+1})$,
and for $1\le p\le n$  
$$
h_{\gamma_p}(x_1,\dots,x_{n-1})=
\int h_{1,\gamma_p}(x_1,\dots,x_n)\overline{h_2(x_n)}G_{j_p,j'_1}(\,dx_n)
$$
with
$h_{1,\gamma_p}(x_1,\dots,x_n)
=h_1(x_{\pi_p(1)},\dots,x_{\pi_p(n)})$,
where $\pi_p(k)=k$ if $1\le k\le p-1$, $\pi_p(p)=n$,
and $\pi_p(k)=k-1$ if $p+1\le k\le n$.

To make the definition of formula (\ref{6.18}) complete I remark that
for $p=1$ we put $\prod\limits_{s=1}^0Z_{G,{j_s}}(\,dx_s)\equiv1$ and for
$p=n$  $\prod\limits_{s=n}^{n-1} Z_{G,{j_s}}(\,dx_s)\equiv1$.}

\medskip\noindent
{\it Proof of the Corollary.}\/ We get the result of the corollary 
by applying Theorem~6.1 in the special case when the second  
Wiener--It\^o integral is defined by formula~(\ref{6.17}) 
instead of~(\ref{6.2}). We have to check that in this case 
the function $h_{\gamma_0}$ corresponding to the diagram 
$\gamma_0$ agrees with the function $h_{\gamma_0}$ defined in 
the corollary, and to calculate the functions $h_{\gamma_p}$ 
defined in~(\ref{6.13}) for the remaining 
diagrams~$\gamma_p$, $1\le p\le n$. In this case 
$\pi_{\gamma_p}(k)=k$ for $1\le k\le p-1$, 
$\pi_{\gamma_p}(k)=k+1$ for $p\le k\le n-1$, 
$\pi_{\gamma_p}(n)=p$, $\pi_{\gamma_p}(n+1)=n+1$, hence
$$
(x(\gamma_p)_1,\dots,x(\gamma_p)_{n+1})
=(x_1,\dots,x_{p-1},x_{p+1},\dots,x_n,x_p,x_{n+1}),
$$
and 
$$
\bar h_{\gamma_p}(x(\gamma_p)_1,\dots,x(\gamma_p)_{n+1})=
h_1(x(\gamma_p)_1,\dots,x(\gamma_p)_n) h_2(-x(\gamma_p)_n)
$$
for $1\le p\le n$. On the other hand, 
$h_2(-x)=\overline{h_2(x)}$, since $h_2\in{\cal K}_{1,j_1'}$.
Thus
\begin{eqnarray*}
&&\bar{\bar h}_{\gamma_p}(x(\gamma_p)_1,\dots,x(\gamma_p)_{n-1}) \\
&&\qquad=
\int h_1(x(\gamma_p)_1,\dots,x(\gamma_p)_{n-1},x(\gamma_p)_n) 
\overline{h_2(x(\gamma_p)_n)}G_{j_p,j'_1}(\,dx(\gamma_p)_n).
\end{eqnarray*}
Then simple calculation shows that for $\gamma=\gamma_p$
the kernel function $h_\gamma=h_{\gamma_p}$ in formula~(\ref{6.15})
agrees with the function~$h_{\gamma_p}$ defined in the corollary
for all $1\le p\le n$, and Theorem~6.1 yields 
identity~(\ref{6.18}) under the conditions of the~corollary.
The corollary is proved.

\medskip
The proof of Theorem~6.1 is similar to the proof of the diagram 
formula (Theorem~5.3 in~\cite{11}). It applies the same method, 
only the notation becomes more complicated than the also rather 
complicated notation of the original proof, since we have to 
work with spectral measures of the form $G_{j_s,j'_t}$ and random 
spectral measures of the form $Z_{G,{j_s}}$ or $Z_{G,{j'_t}}$ instead 
of the  spectral measure $G$ and random spectral measure $Z_G$. 
Hence I decided not to describe the complete proof, I only 
concentrate on its main ideas and the formulas that explain 
why such a formula appears in the diagram formula. The
interested reader can reconstruct the proof by means of a 
careful study of the proof of Theorem~5.3 in~\cite{11}.

\medskip\noindent
{\it A sketch of proof for Theorem 6.1.} The proof of Part~A 
is relatively simple. One can check that the function $h_\gamma$ 
satisfies relation~(a) in the definition of the functions in
${\cal K}_{n+m-2|\gamma|,j_{r_1},\dots,j_{r_{n-|\gamma|}},
j'_{t_1},\dots,j'_{t_{m-|\gamma|}}}$
given in Section~5 by exploiting formula~(\ref{6.13}), the similar 
property of the functions $h_1$ and $h_2$ together with the 
symmetry property $G_{j,j'}(-A)=\overline{G_{j,j'}(A)}$ for all 
$1\le j,j'\le d$ and sets~$A$ of the spectral measure~$G$.

To prove the inequality formulated in Part~A let us first 
rewrite the definition of $h_\gamma$ in~(\ref{6.13}) by replacing all 
measures of the form $G_{j.j'}(dx)$ by 
$g_{j,j'}(x)\mu(\,dx)=G_{j,j'}(\,dx)$, where $\mu$ is a 
dominating measure for all complex measures 
$G_{j,j'}$, $g_{j,j'}$ is the Radon--Nikodym derivative of 
$G_{j,j'}$ with respect to~$\mu$, and observe that the 
inequality~(\ref{3.2}) and formula~(\ref{6.12}) 
and~(\ref{6.13}) imply that
\begin{eqnarray*}
&&|\bar{\bar h}_\gamma(x(\gamma)_1,\dots,x(\gamma)_{n+m-2|\gamma|})| \\
&&
\qquad\le\int 
h_1(x_{\pi_\gamma(1)},\dots,x_{\pi_\gamma(n-|\gamma|)},
x_{\pi_\gamma(n+m-2|\gamma|+1)},\dots,x_{\pi_\gamma(n+m-|\gamma|+1)})  \\
&& 
\qquad\qquad h_2(x_{\pi_\gamma(n-|\gamma|+1)},\dots, 
x_{\pi_\gamma(n+m-2|\gamma|)},  \\ 
&&\qquad\qquad\qquad -x_{\pi_\gamma(n+m-|2\gamma|+1)},\dots, 
-x_{\pi_\gamma(n+m-|\gamma|)})   \\
&&\qquad\qquad\qquad \prod_{k=1}^{|\gamma|} 
\sqrt{g_{j_{v_k},j_{v_k}}(x_{\pi_\gamma(n+m-2|\gamma|+k)})} 
\sqrt{g_{j'_{w_k},j'_{w_k}}(x_{\pi_\gamma(n+m-2|\gamma|+k)})} \\ 
&&\qquad\qquad\qquad\qquad
\mu(\,dx_{\pi_\gamma(n+m-2|\gamma|+k)}).
\end{eqnarray*}
We get by applying the Schwarz inequality, the evenness of the
measures $G_{j,j}$ and by 
replacing the measures of the form $g_{j,j}(x)\mu(\,dx)$ 
or $g_{j',j'}(x)\mu(\,dx)$ by the measures of the form 
$G_{j,j}(\,dx)$ and $G_{j',j'}(\,dx)$ that
\begin{eqnarray*}
&&|\bar{\bar h}_\gamma(x(\gamma)_1,\dots,x(\gamma)_{n+m-2|\gamma|})|^2 \\
&&\qquad\le\int
|h_1(x_{\pi_\gamma(1)},\dots,x_{\pi_\gamma(n-|\gamma|)},
x_{\pi_\gamma(n+m-2|\gamma|+1)},\dots,x_{\pi_\gamma(n+m-|\gamma|+1)})|^2  \\
&&\qquad\qquad\qquad \prod_{k=1}^{|\gamma|}
G_{j_{v_k},j_{v_k}}(\,dx_{\pi_\gamma(n+m-2|\gamma|+k)})\\ 
&&\qquad\qquad \int 
|h_2(x_{\pi_\gamma(n-|\gamma|+1)},\dots, 
x_{\pi_\gamma(n+m-2|\gamma|)},  \\ 
&&\qquad\qquad\qquad\qquad\qquad -x_{\pi_\gamma(n+m-|2\gamma|+1)},\dots, 
-x_{\pi_\gamma(n+m-|\gamma|)})|^2\\
&&\qquad\qquad\qquad\qquad \prod_{k=1}^{|\gamma|}
G_{j'_{w_k,w_k}} (\,dx_{\pi_\gamma(n+m-2|\gamma|+k)}).
\end{eqnarray*}
Let us integrate the last inequality with respect to the
product measure
\begin{eqnarray*} 
&&\prod_{k=1}^{n-|\gamma|}G_{j_{r_k},j_{r_k}}(\,dx(\gamma)_k)
\prod_{l=1}^{m-|\gamma|} G_{j'_{t_l},j'_{t_l}}(\,dx(\gamma)_{n-|\gamma|+l}) \\
&&\qquad=\prod_{k=1}^{n-|\gamma|}G_{j_{r_k},j_{r_k}}(\,dx_{\pi_\gamma(k)})
\prod_{l=1}^{m-|\gamma|}G_{j'_{t_l},j'_{t_l}}(\,dx_{\pi_\gamma(n-|\gamma|+l)}).
\end{eqnarray*}
A careful analysis shows that the inequality we get in such a
way agrees with the inequality formulated in Part~A of Theorem~6.1.
Indeed, we get at the left-hand side of this inequality 
$\|\bar{\bar h}_\gamma\|$ with the norm formulated in Part~A of 
Theorem~6.1, and the right-hand side equals the product 
$\|h_1\|\|h_2\|$. We got the same integrals as the integrals 
defining these norms, only we integrate by the variables of 
the functions $h_1$ and $h_2$ in a different order. We also 
have to exploit that the measures $G_{j,j}$ are symmetric, 
hence the value of the integrals we are investigating does 
not change if we replace the coordinate $x_k$ by $-x_k$ in 
the kernel function for certain coordinates $k$.

Next I turn to the proof of Part~B of Theorem~6.1. First we 
prove this result, i.e. identity~(\ref{6.16}) in the special 
case when both $h_1$ and $h_2$ are simple functions. We may 
also assume that they are adapted to the same regular system
$$
{\cal D}=\{\Delta_p,\;p=\pm1,\pm2,\dots,\pm N\},
$$ 
and by a possible further division of the sets $\Delta_p$ we may 
also assume that the elements of ${\cal D}$ are very small. More 
explicitly, first we choose such a measure $\mu$ on ${\mathbb R}^\nu$ which 
has finite value on all compact sets, all complex  
measures $G_{k,l}$, $1\le k,l\le d$ are absolutely continuous with 
respect to $\mu$, and their Radon--Nikodym derivatives satisfy 
the inequality $|\frac{dG_{k,l}}{d\mu}(x)|\le1$ for all 
$x\in {\mathbb R}^\nu$. Fix a small number~$\varepsilon>0$. We may achieve, 
by splitting up the sets $\Delta_p$ into smaller sets if it is 
necessary, that $\mu(\Delta_p)\le\varepsilon$ for all 
$\Delta_p\in{\cal D}$.
Let us fix a number $u_p\in\Delta_p$ in all sets 
$\Delta_p\in{\cal D}$. We can express the product 
$I_n(h_1|j_1,\dots,j_n)I_m(h_2|j_1',\dots,j'_m)$ as
\begin{eqnarray*}
&& \!\!\!\!\!\!\!\!\!\!
I=I_n(h_1|j_1,\dots,j_n)I_m(h_2|j_1',\dots,j'_m)
={\sum}' h_1(u_{p_1},\dots,u_{p_n})h_2(u_{q_1},\dots,u_{q_m})\\
&&\qquad\qquad Z_{G,{j_1}}(\Delta_{p_1})\cdots Z_{G,{j_n}}(\Delta_{p_n})
Z_{G,{j'_1}}(\Delta_{q_1})\cdots Z_{G,{j'_m}}(\Delta_{q_m}).
\end{eqnarray*}
The summation in the sum $\sum'$ goes through all pairs 
$((p_1,\dots,p_n),(q_1,\dots,q_m))$ such that 
$p_k,\,q_l\in\{\pm1,\dots,\pm N\}$, \ $k=1,\dots,n$, $l=1,\dots,m$,
and $p_k\neq\pm p_{\bar k}$, if $k\neq \bar k$, and 
$q_l\neq\pm q_{\bar l}$ if $l\neq\bar l$.

Write
\begin{eqnarray*}
I&=&\sum_{\gamma\in\Gamma}
{\sum}^\gamma \,
h_1(u_{p_1},\dots,u_{p_n}) h_2(u_{q_1},\dots,u_{q_m})  \\
&&\qquad\qquad Z_{G,{j_1}}(\Delta_{p_1})\cdots Z_{G,{j_n}}(\Delta_{p_n})
Z_{G,{j'_1}}(\Delta_{q_1})\cdots Z_{G,{j'_n}}(\Delta_{q_m}).
\end{eqnarray*}
where $\sum^\gamma$ contains those terms of $\sum'$ for which
$p_k=q_l$ or $p_k=-q_l$ if the vertices $(1,k)$ and $(2,l)$ are
connected in $\gamma$, and $p_k\neq \pm q_l$ if $(1,k)$ and $(2,l)$
are not connected in~$\gamma$.

Let us introduce the notation
\begin{eqnarray*}
\Sigma^\gamma&=&
{\sum}^\gamma \,
h_1(u_{p_1},\dots,u_{p_n}) h_2(u_{q_1},\dots,u_{q_m})  \\
&&\qquad\qquad Z_{G,{j_1}}(\Delta_{p_1})\cdots Z_{G,{j_n}}(\Delta_{p_n})
Z_{G,{j'_1}}(\Delta_{q_1})\cdots Z_{G,{j'_n}}(\Delta_{q_m}).
\end{eqnarray*}
for all $\gamma\in\Gamma$.

We want to show that for small $\varepsilon>0$ (where $\varepsilon$
is an upper bound for the measure $\mu$ of the sets 
$D_p\in{\cal D}$) the expression $\Sigma^\gamma$ is very close to
\begin{equation}
I_\gamma=I_{n+m-2|\gamma|}(\bar{\bar h}_\gamma|j_{v_1},\dots,j_{v_{(n-|\gamma|}},
j'_{w_1},\dots,j'_{w_{m-|\gamma|}})  \label{6.19}
\end{equation}
for all $\gamma\in\Gamma$. For this goal we make the decomposition
$\Sigma^\gamma=\Sigma^\gamma_1+\Sigma^\gamma_2$ of $\Sigma^\gamma$ with
\begin{eqnarray*}
\Sigma_1^\gamma&=&{\sum}^\gamma\, h_1(u_{p_1},\dots,u_{p_n})
h_2(u_{q_1},\dots,u_{q_m})
\prod_{k\in A_1}Z_{G,{j_k}}(\Delta_{p_k})
\prod_{l\in A_2} Z_{G,{j'_l}}(\Delta_{q_l})\\
&&\qquad\qquad\qquad\cdot
\prod_{(k,l)\in B} 
E\left(Z_{G,{j_k}}(\Delta_{p_k})Z_{G,{j'_l}}(\Delta_{q_l})\right)
\end{eqnarray*}
and
$$
\Sigma^\gamma_2=\Sigma^\gamma-\Sigma^\gamma_1,
$$
where the sets $A_1$, $A_2$ and $B$ were defined in 
formulas~(\ref{6.6}), (\ref{6.7}) and~(\ref{6.8}).

It is not difficult to check that both $\Sigma^\gamma_1$ and 
$\Sigma^\gamma_2$ are real valued random variables. We want to
show that $\Sigma^\gamma_1$ is close to the random variable 
$I_\gamma$ introduced in~(\ref{6.19}) while $\Sigma^\gamma_2$ is a 
small error term. To understand the behaviour of 
$\Sigma^\gamma_1$ observe that
$$
E(Z_{G,{j_k}}(\Delta_{p_k})Z_{G,{j'_l}}(\Delta_{q_l})
=E(Z_{G,{j_k}}(\Delta_{p_k})\overline{Z_{G,{j'_l}}(-\Delta_{q_l})}=0
$$
if $\Delta_{p_k}=\Delta_{q_l}$
(and as a consequence
if $\Delta_{p_k}\cap(-\Delta_{q_l})=\emptyset$), and
$$
E(Z_{G,{j_k}}(\Delta_{p_k})Z_{G,{j'_l}}(\Delta_{q_l})
=E(Z_{G,{j_k}}(\Delta_{p_k})\overline{Z_{G,{j'_l}}(-\Delta_{q_l})}
=G_{j_k,j'_l}(\Delta_{p_k})
$$
if $\Delta_{p_k}=-\Delta_{q_l}$. In the case $(k,l)\in B$ one 
of these possibilities happens.

These relations make possible to rewrite $\Sigma^\gamma_1$ 
in a simpler form. It can be rewritten in the form of a 
Wiener--It\^o integral of order $n+m-2|\gamma|$ with 
integration with respect to the  random measure 
$\prod\limits_{k\in A_1}Z_{G,{j_k}}(\,dx_k)
\prod\limits_{l\in A_2} Z_{G,{j'_l}}(\,dx_l)$.
Then we can rewrite this integral, by reindexing its 
variables in a right way to an integral very similar 
to the Wiener--It\^o integral~(\ref{6.14}) (with the same 
parameter $\gamma$). The difference between these 
two expressions is that the kernel function $h'_\gamma$ 
of the Wiener--It\^o integral expressing 
$\Sigma^\gamma_1$ is slightly different from the kernel 
function $\bar{\bar h}_\gamma$ appearing in the other integral. 
The main difference between these two kernel functions
is that there is a small set in the domain of 
integration where $h'_\gamma$ disappears, while 
$\bar{\bar h}_\gamma$ may not disappear. But the two 
Wiener--It\^o integrals are very close to each other. An 
adaptation of the argument in the proof of Theorem~5.3 
in~\cite{11} shows that
$$
E(\Sigma_1^\gamma-I_\gamma)^2\le C\varepsilon
$$
with an appropriate constant $C>0$.

We also want to show that $\Sigma^\gamma_2$ is a 
negligibly small error term. To get a good upper 
bound on $E(\Sigma^\gamma_2)^2$ we write it in the form
\begin{eqnarray*}
E(\Sigma_2^\gamma)^2&=&{\sum}^{\gamma}_2\, 
h_1(u_{p_1},\dots,u_{p_n}) h_2(u_{q_1},\dots,u_{q_m}) \\
&&\qquad\qquad h_1(u_{\bar p_1},\dots,u_{\bar p_n}) 
h_2(u_{\bar q_1},\dots,u_{\bar q_m}) \\
&&\qquad\qquad\qquad \Sigma^\gamma_3(p_k,q_l,p_{\bar k},q_{\bar l},
\;k,\bar k\in\{1,\dots,n\},
\; l,\bar l\in\{1,\dots,m\}) 
\end{eqnarray*}
with
\begin{eqnarray*}
&& \!\!\!\!\!\!\!\!\!
\Sigma^\gamma_3(p_k,q_l,p_{\bar k},q_{\bar l},
\;k,\bar k\in\{1,\dots,n\},
\; l,\bar l\in\{1,\dots,m\}) \\
&&=E\Biggl(\left(\prod_{k\in A_1}Z_{G,{j_k}}(\Delta_{p_k})
\prod_{l\in A_2}Z_{G,{j'_l}}(\Delta_{q_l})
\prod_{\bar k\in A_1}Z_{G,{j_{\bar k}}}(\Delta_{p_{\bar k}})
\prod_{\bar l\in A_2}Z_{G,{j'_{\bar l}}}(\Delta_{q_{\bar l}})\right)\  \\
&&\qquad\left[\prod_{(k,l)\in B} 
Z_{G,{j_k}}(\Delta_{p_k})Z_{G,{j'_l}}(\Delta_{q_l})
-E\prod_{(k,l)\in B} Z_{G,{j_k}}(\Delta_{p_k})Z_{G,{j'_l}}(\Delta_{q_l})\right] \\
&&\qquad\left[\prod_{(\bar k,\bar l)\in B} 
Z_{G,{j_{\bar k}}}(\Delta_{p_{\bar k}}) Z_{G,{j'_{\bar l}}}(\Delta_{q_{\bar l}})
-E\prod_{(\bar k,\bar l)\in B} Z_{G,{j_{\bar k}}}(\Delta_{p_{\bar k}})
Z_{G,{j'_{\bar l}}}(\Delta_{q_{\bar l}})\right]\Biggr), 
\end{eqnarray*}
where we sum up in ${\sum}^\gamma_2$ for such sequences of indices 
$p_k,\,q_l,\,p_{\bar k},\,q_{\bar l}$,  
$k,\bar k\in\{1,\dots,n\}$, $l,\bar l\in\{1,\dots,m\}$,
$p_k,p_{\bar k},q_l,q_{\bar l}\in\{\pm1,\dots,\pm N\}$
which satisfy the following properties. For all indices 
$k,l,\bar k$ and $\bar l$,  $p_k=q_l$ or $p_k=-q_l$ if 
$(k,l)\in B$, and similarly $p_{\bar k}=q_{\bar l}$ or 
$p_{\bar k}=-q_{\bar l}$ if $(\bar k,\bar l)\in B$. 
Otherwise all numbers $\pm p_k$ and $\pm q_l$ are
different, and similarly otherwise all $\pm p_{\bar k}$
and $\pm q_{\bar l}$ are different. 

We get a good estimate on $E(\Sigma_2^\gamma)^2$ by giving a good
bound on all terms
\begin{equation}
\Sigma^\gamma_3(p_k,q_l,p_{\bar k},q_{\bar l},
\;k,\bar k\in\{1,\dots,n\},\; l,\bar l\in\{1,\dots,m\}) \label{6.20}
\end{equation}
in the formula expressing it.
This can be done by adapting the corresponding argument in
Theorem~5.3 of~\cite{11}. This argument shows that for most
sets of parameters $p_k,q_k,p_{\bar k},q_{\bar l}$ the term
in~(\ref{6.20}) equals zero. More explicitly, it is
equal to zero if ${\cal A}\neq-\bar{\cal A}$ with
$$
{\cal A}=\{p_k\colon\; k\in A_1\}\cup \{q_l\colon\;l\in A_2\}, \quad
\textrm{and} \quad \bar{\cal A}=\{p_{\bar k}\colon\; 
\bar k\in A_1\}\cup \{q_{\bar l}\colon\;\bar l\in A_2\},
$$
and it also equals zero if ${\cal F}\cup(-{\cal F})$ and 
$\bar {\cal F}\cup(-\bar {\cal F})$ are disjoint, where
$$
{\cal F}=\bigcup_{(k,l)\in B}\{p_k,q_l\} \quad\textrm{and}\quad
\bar{\cal F}=\bigcup_{(\bar k,\bar l)\in B}\{p_{\bar k},q_{\bar l}\}.
$$

These statements can be proved by adapting the corresponding 
argument in Theorem~5.3 of~\cite{11}. More precisely, in the proof 
of the first statement we still need the following additional 
observation. If $(X,Y,Z)$ is a three-dimensional Gaussian 
vector with $EX=EY=EZ=0$, then $EXYZ=0$. (In the proof of 
Theorem~5.3 in~\cite{11} we needed this statement only in a 
special case when it trivially holds.)

To prove this statement let us apply the following orthogonalization
for the random variables $X$, $Y$ and $Z$. Write $Y=\alpha X+\eta$,
$Z=\beta_1 X+\beta_2\eta+\zeta$, where $X,\eta,\zeta$ are orthogonal,
(jointly) Gaussian random variables with expectation zero. Then they
are also independent, hence 
$EXYZ=EX(\alpha X+\eta)(\beta_1X+\beta_2\eta+\zeta)=0$.

In the remaining cases the expression in~(\ref{6.20}) can be estimated 
(again by adapting the argument of Theorem~5.3 in~\cite{11}) in the 
following way.
\begin{eqnarray*}
&&\Sigma^\gamma_3(p_k,q_l,p_{\bar k},q_{\bar l},
\;k,\bar k\in\{1,\dots,n\},\; l,\bar l\in\{1,\dots,m\}) \\
&&\qquad \le C\varepsilon {\prod}'
\mu(\Delta_{p_k})\mu(\Delta_{l_q})
\mu(\Delta_{p_{\bar k}})\mu(\Delta_{q_{\bar l}})
\end{eqnarray*}
with some constant $C$ (not depending on~$\varepsilon$) and the
measure $\mu$ dominating the complex measures 
$G_{j,k}$ with the properties we demanded at the start of 
the proof. The sign~$'$ in the product ${\prod}'$ means that 
first we take the sets
$\Delta_{p_k}$, $\Delta_{q_l}$, $\Delta_{p_{\bar k}}$, $\Delta_{q_{\bar l}}$
for all parameters $k,\bar k\in\{1,\dots,n\}$ and 
$l,\bar l\in\{1,\dots,m\}$, then if a set $\Delta$ appears twice 
in the sequence of these sets we omit one of them. Then if both 
the sets $\Delta$ and $-\Delta$ appear for some set $\Delta$, 
then we omit one of them from this sequence. Then we take 
in~${\prod}'$ the product of the terms~$\mu(\Delta)$ with
the sets~$\Delta$ in the remaining sequence.

It can be proved with the help of the estimates on the terms 
in~(\ref{6.20}) (see again Theorem~5.3 in~\cite{11}) that
$$
E(\Sigma_2^\gamma)^2\le C\varepsilon.
$$
It is not difficult to prove part~B of Theorem~6.1 with the help of 
the estimates on $E(\Sigma_1^\gamma-I_\gamma)^2\le C\varepsilon$ and 
$E(\Sigma_2^\gamma)^2\le C\varepsilon$ if $h_1$ and $h_2$ are simple
functions. One only has to make an appropriate limiting procedure
with $\varepsilon\to0$. Then we can complete the proof of Theorem~6.1
similarly to the proof of Theorem~5.3 in~\cite{11} by means of an 
appropriate approximation of Wiener--It\^o integrals with 
Wiener--It\^o integrals of simple functions. In this approximation 
we have to apply Lemma~5.1 and the properties of the Wiener--It\^o
integrals, in particular the already proved Part~A of Theorem~6.1.

\section{Wick polynomials and their relation to 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{11}) which shows an important relation between 
Wiener--It\^o integrals and Hermite polynomials. Here I shall 
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. It enables us to present the shift 
transforms of a random variable given in the form of a sum of 
multiple Wiener--It\^o integrals in such a way that helps us in 
the study of limit theorems for non-linear functionals of a 
vector valued stationary Gaussian field.

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{11}. 

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 ${\cal H}_1$ and ${\cal H}$. A square 
integrable (real valued) random variable is in ${\cal H}$ 
if and only if it is measurable with respect to the 
$\sigma$-algebra ${\cal B}={\cal B}(X_t,\;t\in T)$, and 
the scalar product in ${\cal H}$ is defined as 
$\langle\xi,\eta\rangle=E\xi\eta$, $\xi,\,\eta\in{\cal H}$. 
The Hilbert space ${\cal H}_1\subset{\cal H}$ is the subspace 
of ${\cal H}$ generated by the finite linear combinations 
$\sum c_jX_{t_j}$, $t_j\in T$, with real coefficients. We
consider only such sets of Gaussian random variables 
$X_t$ for which ${\cal H}_1$ is separable. Otherwise 
$X_t$, $t\in T$, can be arbitrary, but the most interesting 
case for us is when $T={\mathbb Z}^\nu\times\{1,\dots,d\}$, 
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 Gaussian stationary 
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{11}.

\medskip\noindent
{\bf Theorem 7A.} {\it Let $Y_1,Y_2,\dots$ be an orthonormal basis
in the Hilbert space ${\cal H}_1$ defined above with the help of a set
of Gaussian random variables $X_t$, $t\in T$. Then the set of all
possible finite products $H_{j_1}(Y_{l_1})\cdots H_{j_k}(Y_{l_k})$
is a complete orthogonal system in the Hilbert space ${\cal H}$
defined above. (Here, and in the subsequent discussion $H_j(\cdot)$ 
denotes the $j$-th Hermite polynomial with leading coefficient~1.)}

\medskip
Let ${\cal H}_{\le n}\subset{\cal H}$, $n=1,2,\dots$, (with the 
previously introduced Hilbert space ${\cal H}$) denote the linear 
subspace of the Hilbert space ${\cal H}$ 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 ${\cal H}_0={\cal H}_{\le 0}$ 
consist of the constant functions, and let
${\cal H}_n={\cal H}_{\le n}\ominus{\cal H}_{\le n-1}$, $n=1,2,\dots$, where
$\ominus$ denotes orthogonal completion. It is clear that the Hilbert
space ${\cal H}_1$ given in this definition agrees with the previously
defined Hilbert space ${\cal H}_1$. If $\xi_1,\dots,\xi_m\in{\cal H}_1$,
and $P_n(x_1,\dots,x_m)$ is a polynomial of degree $n$, then
$P_n(\xi_1,\dots,\xi_m)\in{\cal H}_{\le n}$. Then Theorem~7A implies that
\begin{equation}
{\cal H}={\cal H}_0+{\cal H}_1+{\cal H}_2+\cdots,  \label{7.1}
\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 $\xi_1,\dots,\xi_m\in{\cal H}_1$
be given. The Wick polynomial $\colon\!P(\xi_1,\dots,\xi_m)\colon$
determined by them is defined as the orthogonal projection 
of the random variable $P(\xi_1,\dots,\xi_m)$ to the above 
defined subspace ${\cal H}_n$ of the Hilbert space ${\cal H}$.}

\medskip
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 $\xi_1,\dots,\xi_m$, $\xi_j\in{\cal H}_1$ 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 
${\cal H}_1$, 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
$\colon\!P_1(\xi_1,\dots,\xi_m)\colon=\colon\!P(\xi_1,\dots,\xi_m)\colon$, 
i.e. the value of a Wick polynomial does not depend on 
its representation.

\medskip
It is clear that Wick polynomials of different degree are 
orthogonal. Given some $\xi_1,\dots,\xi_m\in{\cal H}_1$ 
define the subspaces
${\cal H}_{\le n}(\xi_1,\dots,\xi_m)\subset{\cal H}_{\le n}$, 
$n=1,2,\dots$, as the set of all polynomials of the random 
variables $\xi_1,\dots,\xi_m$ with degree less than or equal 
to~$n$. Let
${\cal H}_{\le0}(\xi_1,\dots,\xi_m)
={\cal H}_{0}(\xi_1,\dots,\xi_m)={\cal H}_0$,
and 
${\cal H}_n(\xi_1,\dots,\xi_m)={\cal H}_{\le n}(\xi_1,\dots,\xi_m)\ominus
{\cal H}_{\le n-1}(\xi_1,\dots,\xi_m)$. With the help of this notation I
formulate the following result given in~Proposition~2.2 of~\cite{11}.

\medskip\noindent
{\bf Theorem 7B.} {\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 \hfill\break  
${\cal H}_n(\xi_1,\dots,\xi_m)$.}

\medskip 
This result has the following important consequences formulated in
Corollaries~2.3 and~2.4 in~\cite{11}.

\medskip\noindent
{\bf Corollary 7C.} {\it Let $\xi_1,\dots,\xi_m$ be an orthonormal
system in ${\cal H}_1$, and let
$$
P(x_1,\dots,x_m)=\sum c_{j_1,\dots,j_m}x^{j_1}\cdots x_m^{j_m}
$$
be a homogeneous polynomial, i.e. let $j_1+\cdots +j_m=n$ with some
fixed number~$n$ for all sets $(j_1,\dots,j_m)$ appearing in this
summation. Then
$$
\colon\!P(\xi_1,\dots,\xi_m)\colon=\sum
c_{j_1,\dots,j_m}H_{j_1}(\xi_1)\cdots H_{j_m}(\xi_m).
$$
In particular,
$$
\colon\!\xi^n\colon=H_n(\xi) \quad \textrm{if } \xi\in{\cal H}_1, 
\textrm{ and } E\xi^2=1.
$$
}

\medskip\noindent
{\bf Corollary 7D.} {\it Let $\xi_1,\xi_2,\dots$ be an orthonormal
basis in ${\cal H}_1$. Then the random variables
$H_{j_1}(\xi_1)\cdots H_{j_k}(\xi_k)$, $k=1,2,\dots$,
$j_1+\cdots+j_k=n$, form a complete orthogonal basis in ${\cal H}_n$.}

\medskip
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 e.g. in Lemma~5.2 of~\cite{11}.
\begin{equation}
H_n(x)=xH_{n-1}(x)-(n-1)H_{n-2}(x) \quad\textrm{for \ }n=1,2,\dots, 
\label{7.2}
\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 7.1.} {\it Let $U_1,\dots,U_{n+1}$, $n\ge1$, be 
elements in ${{\cal H}_1}$. Then
\begin{eqnarray}
&&\colon\! U_1\cdots U_n\colon U_{n+1} \label{7.3} \\
&&\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 U_{n+1}\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 7.1.}\/ Formula (\ref{7.3}) 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{7.3}) equals $U_1H_n(U_1)$, 
while its right-hand side equals $H_{n+1}(U_1)+nH_{n-1}(U_1)$ by 
Corollary~7C, and these two expressions are equal by formula (\ref{7.2}). 
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{7.3}) 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~7C. A comparison of these expressions together with 
relation~(\ref{7.2}) imply that identity~(\ref{7.3}) holds 
in this case. If $U_{n+1}=V_{k+1}$, then the left-hand side 
of~(\ref{7.3}) 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{7.3}) holds in this case, too.

In the general case we can choose some independent Gaussian 
random variables $Z_1,\dots,Z_m$ in ${\cal H}_1$ 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{7.3}) 
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{7.3}) are multi-linear functionals 
on the $n$-fold direct product 
${\cal H}_1\times\cdots\times{\cal H}_1$, this implies that 
formula~(\ref{7.3}) also holds for the random 
variables~$U_1,\dots,U_{n+1}$. Proposition~7.1 is proved. 

\medskip
We can prove the following multivariate version of It\^o's 
formula with the help of Proposition~7.1 and the diagram 
formula for multiple Wiener--It\^o integrals for 
vector-valued Gaussian stationary fields.

\medskip\noindent
{\bf Theorem 7.2. Multivariate version of It\^o's formula.} 
{\it Let us have some vector valued Gaussian stationary 
random field with a vector valued random spectral measure
$Z_G=(Z_{G,1},\dots,Z_{G,d})$. Let us consider some functions
functions $\varphi_p\in {\cal K}_{1,j_p}$, $1\le p\le n$,
$1\le j_p\le d$, and define with their help the random variables
$U_p=\int\varphi_p(x)Z_{G,{j_p}}(\,dx)\in{\cal H}_1$, $1\le p\le n$. 
The identity
\begin{equation}
\colon\!U_1\cdots U_n\colon=
\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) 
\label{7.4}
\end{equation}
holds.}

\medskip\noindent
{\it Proof of Theorem 7.2.}\/ Relation~(\ref{7.4}) 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 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  
\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) \\
&&\qquad =
\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\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}) \\
&&\qquad\qquad\quad 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}),
\end{eqnarray*}
since formula~(\ref{6.18}) gives this identity with the above choice of
$h_1$ and $h_2$. To see this observe that
\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(x_{n+1})}
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*}
for $1\le p\le n$, since by formula~(\ref{3.7}) 
and the relation $U_{n+1}=\overline{U}_{n+1}$
$$
\int\varphi_p(x)\overline{\varphi_{n+1}(x)}G_{j_p,j_{n+1}}(\,dx)=EU_pU_{n+1},
$$
and
$$
h_{\gamma_0}(x_1,\dots,x_{n+1})=
\varphi_1(x_1)\cdots\varphi_n(x_n) \varphi_{n+1}(x_{n+1}).
$$
This formula 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 this formula with~(\ref{7.3}) we get that
the statement of Theorem~7.2 holds also for $n+1$. Theorem~7.2 is
proved.

\medskip
In Theorem~7.2 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 Wiener--It\^o integrals.
This implies that all Wick polynomials of random variables from
some ${\cal H}_{1,j}$, $1\le j\le d$, can be written in the form of a 
sum of Wiener--It\^o integrals. In the next simple corollary of 
Theorem~7.2  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 
$\varphi_{j,k}(x)\in {\cal K}_{1,j}$, $1\le j\le d$, $1\le k\le m$,
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 $\xi_{j,k}\in{\cal H}_{1,j}$. (We defined the real Hilbert space
${\cal H}_{1,j}$  in the formulation of Lemma~3.2. 
Lemma~3.2 stated that the elements of ${\cal H}_{1,j}$
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 7.2.} {\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 $x_{j_s,k-s}$ by the random variables
$$
\xi_{j_s,k_s}=\int \varphi_{j_s,k_s}(x) Z_{G,j_s}(\,dx)
$$ 
in this polynomial (we assumed that $\varphi_{j,k}\in{\cal K}_{1,j}$), 
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 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 
$$
f_{j_1,k_1,\dots,j_n,k_n}(x_1,\dots,x_n)=
\varphi_{j_1,k_1}(x_1)\cdots\varphi_{j_n,k_n}(x_n)
\in{\cal K}_{n,j_1,\dots,j_n}
$$
and the 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{11} contains a 
version of this result for scalar valued fields.

\medskip\noindent
{\it Proof of the Corollary of Theorem 7.2.}\/ By Theorem~7.2 
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.

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

\medskip\noindent
{\bf Proposition 7.3.} {\it For all $n\ge 1$ and functions 
$f\in{\cal K}_{n,j_1,\dots,j_n}$ with some indices $1\le j_s\le d$, 
$1\le s\le n$, $I_n(f|j_1,\dots,j_n)\in{\cal H}_n$ 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 $f_{j_1,\dots,j_n}\in{\cal K}_{n,j_1,\dots,j_n}$
constitute an everywhere dense linear subspace of ${\cal H}_n$.}

\medskip\noindent
{\it Proof of Proposition 7.3.}\/ We shall prove 
Proposition~7.3 by induction with respect to~$n$. 
By Lemma~3.2 Proposition 7.3 holds for $n=1$. 
Indeed, by this result every random variable of the
form $\xi=\sum_{j=1}^d\xi_j$ with some 
$\xi_j\in{\cal H}_{1,j}$ 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 ${\cal H}_1$.

If the statements of Proposition~7.3 hold for all $m<n$, 
then we can say for one part that 
$I_n(f|j_1,\dots,j_n)\in {\cal H}_{\le n}$, because this 
relation holds if $f$ is a simple function, i.e. if 
$f\in\hat{\cal K}_{n,j_1,\dots,j_n}$, and  since 
$\hat{\cal K}_{n,j_1,\dots,j_n}$ is dense in 
${\cal K}_{n,j_1,\dots,j_n}$, and we defined the Wiener--It\^o 
integral by the extension of a bounded operator in the 
general case, the above property remains valid for general 
functions $f\in{\cal K}_{n,j_1,\dots,j_n}$. Moreover, we know that
$I_n(f|j_1,\dots,j_m)$ is orthogonal to any Wiener--It\^o 
integral of the form $I_m(h|j'_1,\dots,j'_m)$ with $m<n$ 
because of relation~(\ref{5.5}). 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 ${\cal H}_m$ by our inductive 
hypothesis. Hence $I_n(f|j_1,\dots,j_n)$ is orthogonal 
to the whole space ${\cal H}_m$ for all $0\le m\le n-1$, 
and this implies that it is contained in the Hilbert 
subspace ${\cal H}_n$. It follows from the corollary of 
Theorem~7.2 that the sums of Wiener--It\^o integrals 
considered in Proposition~7.3 are dense in ${\cal H}_n$, 
and they constitute a linear subspace. Indeed, this 
corollary implies that all Wick polynomials of order $n$ 
can be expressed as a sum of such integrals, and the Wick 
polynomials of order~$n$ are dense in ${\cal H}_n$.
Proposition~7.3 is proved.

\medskip\noindent
{\it Remark.} In Proposition 7.3 we expressed a dense subset of 
${\cal H}_n$ as a sum of $n$-fold Wiener--It\^o integrals, but 
we did not express all elements of ${\cal H}_n$ in such a form. 
But even this weaker result suffices for our purposes.

In the case of scalar valued stationary random fields the situation
is different. In that case we can express all elements of ${\cal H}_n$
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 $h\in{\cal H}$, which represents the elements
$h\in {\cal H}$ in the form of a sum of multiple 
Wiener--It\^o integrals of different multiplicity. Moreover, 
this result has some useful consequences about the properties 
of this representation.

We cannot prove a similar result in the vector valued case. The
main problem is that while in the case of scalar valued models 
if have a sequence of random variables $h_N\in{\cal H}_n$, 
$N=0,1,2,\dots$, such that $h_N\to h_0$ with some 
$h_0\in{\cal H}_n$ in the norm of ${\cal H}_n$ as $N\to\infty$, 
then these random variables can be expressed as the $n$-fold 
Wiener--It\^o integrals of some functions $k_N\in{\cal K}_n$ 
for which $k_N\to k_0$ in the norm of ${\cal K}_n$, in the 
case of vector valued models we do not have a similar result. 
%There is some hope to give a 
%Wiener--It\^o integral representation of all elements of
%${\cal H}_{n}$ by refining our arguments and enlarging the
%class of functions for which we define the Wiener--It\^o 
%integral, but for the time being we have to be satisfied
%with the weaker result formulated in Proposition~7.3.

\medskip
Next we consider a stationary Gaussian field 
$X(p)=(X_1(p),\dots,X_d(p))$, $p\in{\mathbb Z}^\nu$,
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 $Y\in{\cal H}_n$ which can be represented 
as the $n$-fold Wiener--It\^o integral of some
function $h\in{\cal K}_{n,j_1,\dots,j_n}$, 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{7.5}
\end{equation}
I shall express the shift transforms $T_uY$, $u\in{\mathbb Z}^\nu$, 
of $Y$ given in formula~(\ref{7.5}) in a form which can be considered 
as a Fourier type random integral. 

To do this first I recall the definition of 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{11}. 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 ${\cal H}$ for which $T_uT_v=T_{u+v}$.

In Lemma~3.2 I have shown that each random variable 
$U_j\in{\cal H}_{1,j}$ can be written in the form 
$U_j=\int h(x)Z_{G,j}(\,dx)$ with some function 
$h(x)\in{\cal K}_{1,j}$. 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{7.6}
\end{equation}
with some $h\in {\cal K}_{1,j}$. Indeed, relation~(\ref{7.6}) 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{7.6}) 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 $h\in{\cal K}_{1,j}$.

In Proposition~7.4 I present a similar formula about the shift 
transform of a random variable~$Y$ given by formula~(\ref{7.5}). This 
result is useful in the study of limit theorems related to 
non-linear functionals of a stationary Gaussian field.

\medskip\noindent
{\bf Proposition 7.4 about the representation of shift 
transformations.} {\it Let a vector valued stationary Gaussian 
random field $X(p)=(X_1(p),\dots,X_d(p))$, $p\in{\mathbb Z}^\nu$, 
be given with a vector valued random spectral measure 
$$
Z_G=(Z_{G,1},\dots,Z_{G,d}) \textrm{ such that } 
X_j(p)=\int e^{i(p,x)}Z_{G,j}(\,dx)
$$ 
for all $p\in{\mathbb Z}^\nu$ 
and $1\le j\le d$. Let $Y\in{\cal H}_n$ be  the random variable 
defined in formula~(\ref{7.5}) with the help of this vector valued 
random spectral measure~$Z_G$ and some function 
$h\in{\cal K}_{n,j_1,\dots,j_n}$. 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{7.7}
\end{equation}
for all $u\in{\mathbb Z}^\nu$.
}

\medskip\noindent
{\it Proof of Proposition 7.4.}\/ Formula (\ref{7.7}) holds in the 
special case if $n=1$, and $h(x)\in {\cal K}_{1,j}$, 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{7.6}).

I claim that formula~(\ref{7.7}) also holds in the case when the 
random variable $Y$ is given by formula~(\ref{7.5}) 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 $\varphi_s(x)\in{\cal K}_{1,j_s}$, 
$1\le s\le n$. 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~7.2. 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~7B 
$\colon\!U_1\cdots U_n\colon$ is the orthogonal projection of 
the product $U_1\cdots U_n$ to the Hilbert subspace 
${\cal H}_n(U_1,\dots,U_n)$. 
Similarly, $\colon\! (T_uU_1)\cdots (T_uU_n)\colon$ is the orthogonal 
projection of $(T_uU_1)\cdots (T_uU_n)$ to the Hilbert 
subspace ${\cal H}_n(T_uU_1,\dots,T_uU_n)$. 
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{7.5}) 
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~7.2 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{7.7})
also holds if $Y$ is defined by~(\ref{7.5}) 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 $\varphi_{s,k}\in{\cal K}_{1,j_s}$, $1\le s\le n$.

Since functions of the above form are dense in 
${\cal K}_{n,j_1,\dots,j_n}$, $T_u$ 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 $h\to I_n(h|j_1,\dots,j_n)$ are of bounded norm, it is not 
difficult to see that Proposition~7.4 holds in the general case.
Proposition~7.4 is proved.

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

\medskip\noindent
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{8.1}
\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{8.2}
\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{8.1}) as $N\to\infty$ with 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{6} for scalar 
valued stationary random fields, i.e. in the case $d=1$, 
and we have proved some new kind 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{16}. I do not discuss Taqqu's work, 
because here I am interested  in the question how to 
generalize the method in~\cite{6} to prove limit theorems 
also for non-linear functionals of vector valued 
Gaussian stationary random fields. 

In Theorem~6 of~\cite{1} M. A. Arcones formulated a limit 
theorem about non-linear functionals of vector valued Gaussian 
stationary fields, and he referred in his proof to paper~\cite{6},
which is explained in~\cite{11} in more detail. However, as  I 
explained in the introduction, I do consider Arcones' proof 
satisfactory, and I want to give a correct proof. The goal of
this paper is to work out the tools needed to apply the method
of~\cite{6} in the multivariate case. 

The previous part of this paper was about the discussion of 
the notions and results we need to adopt the method of~\cite{11} 
when we are working with multivariate models. In this 
section I explain how to generalize the method of~\cite{11} to
prove limit theorems  when we are working with functionals 
of vector valued Gaussian stationary fields. I shall give 
the proof of the results formulated by Arcones in paper~\cite{13}. 
Here I work out the tools we need in the proof of these
results. They are a rather direct adaptation of 
some results in~\cite{11} to the multivariate case. 

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
transformation~$T_p$ 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{8.1}) in the form
\begin{equation}
S_N=A_N^{-1}\sum_{p\in B_N}T_pY(0). \label{8.3}
\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 $Y(0)$ be a Wick polynomial of order~$n$ which has the
form
$$
Y(0)=\;\;\colon\!\!\!\!\!\!
 \sum_{\substack{(k_1,\dots,k_d)\\ k_1+\cdots+k_d=n}}
 a_{k_1,\dots,k_d}X_1(0)^{k_1}\cdots X_d(0)^{k_d}\colon
$$
with some real coefficients $a_{k_1,\dots,k_d}$.
Then by the corollary of Theorem~7.2 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
$$
Y(0)= \!\!\! \sum_{\substack{(k_1,\dots,k_d)\\ k_1+\cdots +k_d=n}} \!\!\!
 a_{k_1,\dots,k_d}\int {\mathbb I}_1(x_1)\dots {\mathbb I}_1(x_n)\prod_{j=1}^d
\left(\prod_{t=k_1+\cdots+k_{j-1}+1}^{k_1+\cdots+k_j} Z_{G,j}(\,dx_t)\right),
$$
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~7.4 we can write
\begin{equation}
T_pY(0)=\sum_{\substack{(k_1,\dots,k_d)\\ k_1+\cdots +k_d=n}}
 a_{k_1,\dots,k_d}\int e^{i(p,x_1+\cdots+x_n)}\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{8.4}
\end{equation}
for all indices $p\in{\mathbb Z}^\nu$.

We get by summing up formula~(\ref{8.4}) 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_1+\cdots +k_d=n}}
 a_{k_1,\dots,k_d}\int&&\prod _{l=1}^\nu 
\frac{e^{iN(x_1^{(l)}+\cdots+x_n^{(l)})}-1} 
{e^{i(x_1^{(l)}+\cdots+x_n^{(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{8.2}).) 
We can rewrite the above integral with the change of 
variables $y_l=Nx_l$, $1\le l\le n$, in the following form.
\begin{equation}
S_N=
\sum_{\substack{(k_1,\dots,k_d)\\ k_1+\cdots +k_d=n}}
 \int h^N_{k_1,\dots,k_d}(y_1,\dots,y_n)\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{8.5}
\end{equation}
where
$$
h^N_{k_1,\dots,k_d}(y_1,\dots,y_n)=a_{k_1,\dots,k_d}\prod_{l=1}^\nu 
\frac{e^{i(y_1^{(l)}+\cdots+y_n^{(l)})}-1} 
{N(e^{i(y_1^{(l)}+\cdots+y_n^{(l)})/N}-1)} 
$$
is a function on $[-N\pi,N\pi)^\nu$, and 
$Z_{G^{(N)},j}(A)=N^{\nu/n}A_N^{-1/n}Z_{G,j}(\frac AN)$ is defined
for all measurable sets $A\subset [-N\pi,N\pi)^\nu$ and 
$j=1,\dots,d$. Here we use the notation 
$y_s=(y_s^{(1)},\dots,y_s^{(\nu)})$, $1\le s\le n$.  
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 
$$
G^{(N)}_{j,j'}(A)=N^{2\nu/n}A_N^{-2/n}G_{j,j'}(\frac AN), 
\quad 1\le j,j'\le d,
$$ 
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, 
$h^N_{k_1,\dots,k_d}\in{\cal K}_{n,j_1,\dots,j_n}
(G^{(N)}_{j_1,j_1},\dots,G^{(N)}_{j_n,j_n})$ 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 $k_1+\cdots+ k_{s-1}=0$.)

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

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 $(G^{(0)}_{j,j'})$, $1\le j,j'\le d$,
defined on ${\mathbb R}^\nu$ in an appropriate sense, then a limiting 
procedure in formula~(\ref{8.5}) supplies the limit theorem
\begin{eqnarray*}
S_N&\to& S_0 \\
&=&\sum_{\substack{(k_1,\dots,k_d)\\ k_1+\cdots +k_d=n}}
 \int h^0_{k_1,\dots,k_d}(y_1,\dots,y_n)\prod _{j=1}^d 
\left(\prod_{t=k_1+\cdots+k_{j-1}+1}^{k_1+\cdots+k_j} Z_{G^{(0)},j}(\,dy_t)\right),
\end{eqnarray*}
in distribution 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 $(G^{(0)}_{j,j'})$, $1\le j,j'\le d$.

\medskip
Next I explain how to work out a precise method to prove limit
theorems on the basis of the above heuristic argument. In the 
scalar valued case this was done in Lemma~8.3 of \cite{11}, and here
I prove the vector valued variant of this result.

In the formulation of Lemma~8.3 of~\cite{11} we had to introduce 
a generalization of the notion of weak convergence of measures  
when we work with locally finite measures, i.e. with measures 
whose restriction to any compact set is finite. Here I introduce 
a slight generalization of this notion, called vague convergence 
in~\cite{11}, to the case when we are working with complex 
measures of locally finite total variation. (The definition of
complex measures on ${\mathbb R}^\nu$ with locally finite total 
variation was explained in Section~4.)

\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{8.6}
\end{equation}
for all continuous functions~$f$ on ${\mathbb R}^\nu$ with a 
bounded support.}

\medskip
I shall take a sequence of sums of $n$-fold Wiener--It\^o 
integrals, and then I formulate Proposition~8.1 which states
that under some appropriate conditions these random integrals 
have a limit, and this limit is also expressed explicitly in 
Proposition~8.1. Besides the representation of non-linear 
functionals of vector valued Gaussian stationary fields by 
means of Wiener--It\^o integrals this is our main tool to 
prove the limit theorems with non-Gaussian limit for 
non-linear functionals of vector valued Gaussian 
stationary 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 
$$
h^N_{j_1,\dots,j_n}(x_1,\dots,x_n)\in {\cal K}_{n,j_1,\dots,j_n}
={\cal K}_{n,j_1,\dots,j_n}(G^{(N)}_{j_1,j_1},\dots,G^{(N)}_{j_n,j_n})
$$
on the torus $[-A_N\pi,A_N\pi)^\nu$ for all $1\le j_k\le d$, 
$1\le k\le n$, 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_n)\\ 
1\le j_k\le d \textrm{ for all }1\le k\le n}} \!\!
\int h^N_{j_1,\dots,j_n}(x_1,\dots,x_n)Z_{G^{(N)},{j_1}}(\,dx_1)
\dots Z_{G^{(N)},{j_n}}(\,dx_n),    \label{8.7}
\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 8.1.} {\it Let us consider the sums of
$n$-fold Wiener--It\^o integrals $Z_N$ defined in 
formula~(\ref{8.7}) 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 
$$
h^N_{j_1,\dots,j_n}(x_1,\dots,x_n)\in 
{\cal K}_{n,j_1,\dots,j_n}(G^{(N)}_{j_1,j_1},\dots,G^{(N)}_{j_n,j_n}).
$$
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 $h^0_{j_1,\dots,j_n}$
for all $1\le j_k\le d$, $1\le k\le n$, such that 
these functions and matrix valued spectral measures 
satisfy the following conditions.

\medskip
\begin{description}
\item[{\rm(a)}] The functions $h^0_{j_1,\dots,j_n}(x_1,\dots,x_n)$ 
are continuous on ${\mathbb R}^{n\nu}$ for all $1\le j_k\le d$, 
$1\le k\le n$,  and for all $T>0$ and indices 
$1\le j_k\le d$, $1\le k\le n$, the functions 
$h^N_{j_1,\dots,j_n}(x_1,\dots,x_n)$ converge uniformly to 
the function $h^0_{j_1,\dots,j_n}(x_1,\dots,x_n)$ on the 
cube $[-T,T]^{n\nu}$ 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}^{n\nu}\setminus [-T,T]^{n\nu}} |h^N_{j_1,\dots,j_n}(x_1,\dots,x_n)|^2
G^{(N)}_{j_1,j_1}(\,dx_1)\dots G^{(N)}_{j_n,j_n}(dx_n)
<\varepsilon^2 \label{8.8} 
\end{equation}
for all $1\le j_k\le d$, $1\le k\le n$, and $N=1,2\dots$ if
$T>T_0$.
\end{description}

\medskip
Then inequality~(\ref{8.8}) holds also for $N=0$, 
\begin{equation}
h^0_{j_1,\dots,j_n}\in{\cal K}_{n,j_1,\dots,j_n}
={\cal K}_{n,j_1,\dots,j_n}(G^{(0)}_{j_1,j_1},\dots G^{(0)}_{j_n,j_n}),
\label{8.9}
\end{equation}
the sum of random integrals  
\begin{equation}
Z_0=\sum_{\substack{(j_1,\dots,j_n)\\ 
1\le j_k\le d, \textrm{ for all }1\le k\le n}}
\int h^0_{j_1,\dots,j_n}(x_1,\dots,x_n)Z_{G^{(0)},{j_1}}(\,dx_1)
\dots Z_{G^{(0)},{j_n}}(\,dx_n) \label{8.10}
\end{equation}  
exists, and the random variables $Z_N$ defined in~(\ref{8.7})
satisfy the relation $Z_N\stackrel{\cal D}{\rightarrow} Z_0$ as 
$N\to\infty$, where $\stackrel{\cal D}{\rightarrow}$ 
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{8.6}) 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.}\/ Proposition~8.1 together with the previous
consideration suggest when we can expect the appearance of a 
non-central limit theorems for non-linear functionals of 
vector valued stationary Gaussian random fields and how to 
prove such a result. 

To prove such limit theorems first we present the non-linear 
functionals we are working with in the form of a finite sum 
of multiple Wiener--It\^o integrals and then we rewrite them 
with an appropriate rescaling. We try to find such a rescaling 
where the random spectral measures which appear in the random 
integrals correspond to such spectral measures which have a 
(vague) limit, and also the kernel functions in these integrals 
have a nice limit satisfying condition~(a) of Proposition~8.1. 
We can prove a non-central limit theorem if the functions and 
measures appearing in the definition of this multiple 
Wiener--It\^o integrals satisfy a compactness type condition 
formulated in condition~(b) of Proposition~8.1. This condition 
plays a very important role in this result. If it does not 
hold, then a different situation arises. In such cases we can 
get a central limit theorem with standard normalization, see 
e.g. Theorem~4 in~\cite{1}, \cite{3} or the discussion of
this problem in a more general setting in the book \cite{14}.
It may be worth mentioning also the paper~\cite{9}, where 
Ho, H.~C. and Sun, T.~C. proved an interesting result about 
the limit distribution of a linear functional of a stationary 
Gaussian process into ${\mathbb R}^2$, where the first 
coordinate of the limit was Gaussian and the second 
coordinate was non-Gaussian. It may be interesting to find 
the natural multivariate generalization of this result.

\medskip
I shall adapt the proof of Lemma~6.6 in~\cite{12} to the 
multivariate case. This is a simplified version of the proof 
of Lemma~8.3 in~\cite{11}. The latter result is actually a slightly 
more general scalar valued version of Proposition~8.1. We 
could have proved also the generalization of Lemma~8.3 in~\cite{11} 
to the multivariate case. But the formulation of Proposition~8.1 
in the present form is sufficient for our purposes.

\medskip\noindent
{\it Proof of Proposition 8.1.}\/ First I show that 
relation~(\ref{8.8}) holds also for $N=0$. To see this let
us first show that the measures $\mu^{(N)}_{j_1,\dots,j_n}$,
$N=1,2,\dots$, defined as
$$
\mu^{(N)}_{j_1,\dots,j_n}(A)=\int_A |h^N_{j_1,\dots,j_n}(x_1,\dots,x_n)|^2
G^{(N)}_{j_1,j_1}(\,dx_1)\dots G^{(N)}_{j_n,j_n}(dx_n), \quad A\subset R^{n\nu}, 
$$
converge vaguely to the locally finite measure $\mu^{(0)}_{j_1,\dots,j_n}$
defined as
$$
\mu^{(0)}_{j_1,\dots,j_n}(A)
=\int_A |h^0_{j_1,\dots,j_n}(x_1,\dots,x_n)|^2
G^{(0)}_{j_1,j_1}(\,dx_1)\dots G^{(0)}_{j_n,j_n}(dx_n), \quad A\subset R^{n\nu}, 
$$
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 $|h^N_{j_1,\dots,j_n}(x_1,\dots,x_n)|^2$ by
$|h^0_{j_1,\dots,j_n}(x_1,\dots,x_n)|^2$ in the definition of the
measures $\mu^{(N)}_{j_1,\dots,j_n}$. Then condition~(a) of 
Proposition~8.1 implies that this relation also holds with the
original definition of the measures $\mu^{(N)}_{j_1,\dots,j_n}$. 

Next I state that the measure $\mu^{(0)}_{j_1,\dots,j_n}$ is finite, 
and the measures $\mu^{(N)}_{j_1,\dots,j_n}$ converge to it not only 
vaguely but also weakly. Indeed, condition~(b) implies that the
sequence of measures $\mu^{(N)}_{j_1,\dots,j_n}$ 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 these sub-subsequences can be 
only its limit with respect to the vague convergence, i.e. it is 
$\mu^{(0)}_{j_1,\dots,j_n}$. This implies that $\mu^{(0)}_{j_1,\dots,j_n}$
is a finite measure, and the sequence of measures
$\mu^{(N)}_{j_1,\dots,j_n}$ converges also weakly to it.

Finally the properties of the functions $h^N_{j_1,\dots,j_n}$,
and their convergence to $h^0_{j_1,\dots,j_n}$ formulated in
condition~(a) imply that also the symmetry property
$h^0_{j_1,,\dots,j_n}(-x_1,\dots,-x_n)
=\overline{h^0_{j_1,,\dots,j_n}(x_1,\dots,x_n)}$ holds, hence
relation (\ref{8.9}) is valid, and the random integral $Z_0$
defined in (\ref{8.10}) is meaningful.

Then I reduce the proof of the relation 
$Z_N\stackrel{\cal D}{\rightarrow} Z_0$ to the proof of the 
following statement. 

Under the conditions of Proposition~8.1
\begin{eqnarray}
&&\sum_{\substack{(j_1,\dots,j_n)\\ 
1\le j_k\le d, \textrm{ for all }1\le k\le d}} \!\!
\int h^0_{j_1,\dots,j_n}(x_1,\dots,x_n) \chi_T(x_1,\dots,x_n )
\label{8.11}  \\
&& \qquad\qquad\qquad\qquad\qquad\qquad
Z_{G^{(N)},{j_1}}(\,dx_1)\dots Z_{G^{(N)},{j_n}}(\,dx_n)  \nonumber \\
&&\qquad\qquad \stackrel{\cal D}{\rightarrow}
\sum_{\substack{(j_1,\dots,j_n) \nonumber \\ 
1\le j_k\le d, \textrm{ for all }1\le k\le d}}
\int h^0_{j_1,\dots,j_n}(x_1,\dots,x_n) \chi_T(x_1,\dots,x_n) \\
&&\qquad\qquad\qquad\qquad\qquad\qquad\qquad 
Z_{G^{(0)},{j_1}}(\,dx_1)\dots Z_{G^{(0)},{j_n}}(\,dx_n), \nonumber 
\end{eqnarray}
as $N\to\infty$, where $\chi_T(x_1,\dots,x_n)$ is the indicator
function of the cube $[-T,T]^{n\nu}$. We state formula~(\ref{8.11}) 
for all such $T>0$ for which the boundary of the cube 
$[-T,T]^{n\nu}$ has zero measure with respect to the measure 
$\underbrace{\mu_0\times\cdots\times\mu_0}_{n \textrm{ times}}$,
where $\mu_0$ is the dominating measure of the complex
measures $G^{(0)}_{j,j'}$, $1\le j,j'\le d$, with locally 
finite total variation which appears in the definition of 
semidefinite matrix valued even measures introduced in 
Section~4 if this definition is applied for the matrix 
valued spectral measure $(G^{(0)}_{j,j'})$.

To prove this reduction let us observe that by formulas~(\ref{5.6}) 
and~(\ref{8.8}) 
\begin{eqnarray*}
&&\!\!\!\!\!\!\!\!\!
E\biggl[\int [1-\chi_T(x_1,\dots,x_n)]h^N_{j_1,\dots,j_n}(x_1,\dots,x_n)\\
&& \qquad\qquad\qquad\qquad 
Z_{G^{(N)},{j_1}}(\,dx_1)\dots Z_{G^{(N)},{j_n}}(\,dx_n)\biggr]^2 \\
&&\!\!\!\!\!\! \le n!\int_{{\mathbb R}^{k\nu}\setminus [-T,T]^{n\nu}}
|h^N_{j_1,\dots,j_n}(x_1,\dots,x_n)|^2
G^{(N)}_{j_1,j_1}(\,dx_1)\dots G^{(N)}_{j_n,j_n}(\,dx_n)<n!\varepsilon^2
\end{eqnarray*}
for all sequences $(j_1,\dots,j_n)$, $1\le j_k\le d$, $1\le k\le n$,
and $N=0,1,2,\dots$ if $T>T_0(\varepsilon)$.  Hence
\begin{eqnarray}
E\biggr[&&\sum_{\substack{(j_1,\dots,j_n) \nonumber \\ 
1\le j_k\le d \textrm{ for all }1\le k\le n}}
\int [1-\chi_T(x_1,\dots,x_n)]h^N_{j_1,\dots,j_n}(x_1,\dots,x_n)\\
&&\qquad\qquad
Z_{G^{(N)},{j_1}}(\,dx_1)\dots Z_{G^{(N)},{j_n}}(\,dx_n)\biggr]^2 
\le d^nn!\varepsilon^2 \label{8.12}
\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 
$h^N_{j_1,\dots,j_n}\to h^0_{j_1,\dots,j_n}$ on any cube $[-T,T]^{n\nu}$ 
we have
\begin{eqnarray*}
&& E\biggl[\int[h^N_{j_1,\dots,j_n}(x_1,\dots,x_n)-h^0_{j_1,\dots,j_n}
(x_1,\dots,x_n)]\chi_T(x_1,\dots,x_n) \\
&&\qquad\qquad\qquad\qquad\qquad\qquad\qquad 
Z_{G^{(N)},{j_1}}(\,dx_1)\dots Z_{G^{(N)},{j_n}}(\,dx_n)\biggr]^2   \\
&&\qquad\le n!\int_{[-T,T]^{n\nu}}
|h^N_{j_1,\dots,j_n}(x_1,\dots,x_n)-h^0_{j_1,\dots,j_n}(x_1,\dots,x_n)|^2\\
&&\qquad\qquad\qquad\qquad\qquad\qquad\qquad 
G^{(N)}_{j_1,j_1}(\,dx_1)\dots G^{(N)}_{j_n,j_n}(\,dx_n)<\varepsilon^2 
\end{eqnarray*}
for all $T>0$ and $(j_1,\dots,j_n)$, $1\le j_k\le d$, $1\le k\le n$, 
if  $N>N_1$ with some $N_1=N_1(T,\varepsilon)$. Hence
\begin{eqnarray}
E\biggr[&&\sum_{1\le j_1,\dots,j_n\le d}
\int[h^N_{j_1,\dots,j_n}(x_1,\dots,x_n)-h^0_{j_1,\dots,j_n}
(x_1,\dots,x_n)]
\label{8.13} \\
&&\qquad
\chi_T(x_1,\dots,x_n)   
Z_{G^{(N)},{j_1}}(\,dx_1)\dots Z_{G^{(N)},{j_n}}(\,dx_n)\biggr]^2
\le d^n\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_n\le d}
\int &&h^N_{j_1,\dots,j_n}(x_1,\dots,x_n)\chi_T(x_1,\dots,x_n) \\
&&\qquad Z_{G^{(N)},{j_1}}(\,dx_1)\dots Z_{G^{(N)},{j_n}}(\,dx_n),
\end{eqnarray*}
and
\begin{eqnarray*}
V_N=V_N(T)=\sum_{1\le j_1,\dots,j_n\le d}
\int &&h^0_{j_1,\dots,j_n}(x_1,\dots,x_n)\chi_T(x_1,\dots,x_n) \\
&&\qquad Z_{G^{(N)},{j_1}}(\,dx_1)\dots Z_{G^{(N)},{j_n}}(\,dx_n),
\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. We can reduce the proof 
of the relation $Z_N\stackrel{\cal D}{\rightarrow}Z_0$ 
to formula~(\ref{8.11}) in the following way. By formula~(\ref{8.12}) 
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^nn!)^{1/2}\varepsilon. 
\end{eqnarray*}
for all $t\in {\mathbb R}^1$ with the random variable $Z_N$ defined in~(\ref{8.7}) 
if $T>T_0$ and $N>N_0(\varepsilon)$. Similarly, 
$|E(e^{itU_N}-e^{itV_N})|
\le |t| (E(U_N-V_N)^2)^{1/2}\le |t|d^{n/2}\varepsilon$
for all $t\in {\mathbb R}^1$ and $N>N_0$ by inequality~(\ref{8.13}).
Finally, $Ee^{itV_N}\to Ee^{itZ_0}$ for all $t\in {\mathbb R}^1$ with $Z_0$
defined in~(\ref{8.10}) if relation (\ref{8.11}) 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
$Z_N\stackrel{\cal D}{\rightarrow}Z_0$. (In formula~(\ref{8.11})
we imposed a condition on the parameter $T>0$. We demanded that 
the boundary of $[-T,T]^{n\nu}$ 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{8.11}) 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 
$[-T,T]^{n\nu}$ has zero measure with respect to the measure 
$\underbrace{\mu_0\times\cdots\times\mu_0}_{n\textrm{ times}}$.
Observe that 
$$
h^0_{j_1,\dots,j_n}(x_1,\dots,x_n)\chi_T(x_1,\dots,x_n)
\in{\cal K}_{n,j_1,\dots,j_n} (G^{(N)}_{j_1,j_1},\dots,G^{(N)}_{j_n,j_n})
$$
for all $T>0$ and $N=0,1,2,\dots$. I claim that for all 
$\varepsilon>0$ and $(j_1,\dots,j_n)$, $1\le j_k\le d$, 
$1\le k\le n$, there is a simple function 
$$
f^\varepsilon_{j_1,\dots,j_n}\in\hat{\cal K}_{n,j_1,\dots,j_n}
(G^{(0)}_{j_1,j_1},\dots,G^{(0)}_{j_n,j_n})
$$
such that it is adapted to such a regular system
${\cal D}=\{\Delta_k,\;k=\pm1,\dots,\pm M\}$ whose 
elements have boundaries with zero $\mu_0$ measure, i.e. 
$\mu_0(\partial\Delta_k)=0$ for all $1\le |k|\le M$,
$\Delta_k\subset [-T,T]^\nu$ for all $1\le |k|\le M$, 
(we choose a regular system ${\cal D}$ in such a way
that all functions $f^\varepsilon_{j_1,\dots,j_n}$ with a 
fixed parameter $\varepsilon>0$ are adapted to it), 
and the functions $f^\varepsilon_{j_1,\dots,j_n}$ satisfy 
the following inequalities.
\begin{eqnarray}
&&\int|h^0_{j_1,\dots,j_n}(x_1,\dots,x_n)\chi_T(x_1,\dots,x_n)
-f^\varepsilon_{j_1,\dots,j_n}(x_1,\dots,x_n)|^2  \nonumber \\
&&\qquad\qquad\qquad\qquad 
G^{(0)}_{j_1,j_1}(\,dx_1)\dots G^{(0)}_{j_n,j_n}(\,dx_n)
<\varepsilon^2 \label{8.14} 
\end{eqnarray}
for all $1\le j_k\le d$, $1\le k\le n$, and also the inequalities
\begin{eqnarray}
&&\int|h^0_{j_1,\dots,j_n}(x_1,\dots,x_n)\chi_T(x_1,\dots,x_n)
-f^\varepsilon_{j_1,\dots,j_n}(x_1,\dots,x_n)|^2  \nonumber\\
&&\qquad\qquad\qquad\qquad 
G^{(N)}_{j_1,j_1}(\,dx_1)\dots G^{(N)}_{j_n,j_n}(\,dx_n)
<\varepsilon^2 \label{8.15} 
\end{eqnarray}
hold for all $1\le j_k\le d$, $1\le k\le n$, and $N\ge N_0$
with some $N_0=N_0(\varepsilon,T)$.

I also claim that
\begin{equation}
Y_N\stackrel{\cal D}{\rightarrow} Y_0
\label{8.16}
\end{equation}
as $N\to\infty$, where
\begin{eqnarray*}
Y_N&=&Y_N(\varepsilon,T) \\
&=& \!\!\!\!\!\!
\sum_{\substack{(j_1,\dots,j_n)\\ 
1\le j_k\le d \textrm{ for all }1\le k\le n}} \!\!\!\!\!
\int f^\varepsilon_{j_1,\dots,j_n}(x_1,\dots,x_n)Z_{G^{(N)},{j_1}}(\,dx_1)
\dots Z_{G^{(N)},{j_n}}(\,dx_n)
\end{eqnarray*}
for $N=0,1,2,\dots$.

Let me remark that the condition that all simple functions 
$f^\varepsilon_{j_1,\dots,j_n}$ (with a fixed parameter $\varepsilon>0$)
are adapted to the same regular system ${\cal D}$ causes no problem.
By an appropriate refinement of those regular systems to which they
are adapted we get such a regular system. Moreover, this refinement
can be done in such a way that the sets $\Delta_k$ in the new
regular system preserve the property that their boundaries have 
zero $\mu_0$ measure.

Relation~(\ref{8.11}) can be proved with the help of relations (\ref{8.14}), 
(\ref{8.15}) and (\ref{8.16}) similarly to the reduction of the relation
$Z_N\stackrel{\cal D}{\rightarrow} Z_0$ to formula~(\ref{8.11}).
Indeed, one gets from inequalities (\ref{8.14}), (\ref{5.6}) and 
the definition of the quantities $V_N$ and $Y_0$, by applying an
argument similar to the proof of relation~(\ref{8.12}) that
$$
E(V_0-Y_0)^2\le n!n^d\varepsilon^2,
$$
and also
$$
E(V_N-Y_N)^2\le n^d n!\varepsilon^2
$$
if $N>N_0(\varepsilon,T)$ by~(\ref{8.15}) and~(\ref{5.6}). 

Then we can show with the help of these relations similarly 
to the reduction of the relation
$Z_N\stackrel{\cal D}{\rightarrow} Z_0$ to formula~(\ref{8.11})
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{8.14}) and (\ref{8.15}), while the second inequality follows
from (\ref{8.16}). 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. $V_N\stackrel{\cal D}{\rightarrow} V_0$ 
as $N\to\infty$, and this is formula~(\ref{8.11}) written with a 
different notation.

It remains to prove~(\ref{8.14}), (\ref{8.15}) 
and (\ref{8.16}). Relation (\ref{8.14}) is a direct 
consequence of Lemma~5.2. Then formula~(\ref{8.15}) 
follows from some classical results about vague 
(and weak) convergence of measures.
Since we are working in the proof of (\ref{8.15}) in a
cube $[-T,T]^{n\nu}$ 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 $G^{(N)}_{j_1,j_1}\times\cdots\times G^{(N)}_{j_n,j_n}$
to the cube $[-T,T]^{n\nu}$ converge weakly to 
the restriction of the product measure 
$G^{(0)}_{j_1,j_1}\times\cdots\times G^{(0)}_{j_n,j_n}$
on the cube $[-T,T]^{n\nu}$, as $N\to \infty$. On the other
hand, the function 
\begin{eqnarray*} 
&& H^0_{j_1,\dots j_n}(x_1,\dots,x_n) \\
&&\qquad= |h^0_{j_1,\dots,j_n}(x_1,\dots,x_n)\chi_T(x_1,\dots,x_n)-
f^\varepsilon_{j_1,\dots,j_n}(x_1,\dots,x_n)|^2
\end{eqnarray*}
is almost everywhere continuous with respect to the
measure $G^{(0)}_{j_1,j_1}\times\cdots\times G^{(0)}_{j_n,j_n}$.
By the general theory about convergence of measures these
properties imply that
\begin{eqnarray*}
&&\int H^0_{j_1,\dots j_n}(x_1,\dots,x_n)
G^{(N)}_{j_1,j_1}(\,dx_1)\dots G^{(N)}_{j_n,j_n}(\,dx_n) \\
&&\qquad \to\int H^0_{j_1,\dots j_n}(x_1,\dots,x_n)
G^{(0)}_{j_1,j_1}(\,dx_1)\dots G^{(0)}_{j_n,j_n}(\,dx_n)
\nonumber
\end{eqnarray*}
as $N\to\infty$. (Such a convergence is proved for 
probability measures e.g. 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{8.14}) imply~(\ref{8.15}).

To prove relation~(\ref{8.16}) 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
$\Delta_k\in{\cal D}$ with the regular system
${\cal D}$ we are working with. (Let me recall
that the boundary of all sets $\Delta_k\in{\cal D}$
has zero $\mu_0$ measure.)

This relation immediately follows if $j=j'$ 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 $\Delta_k\in{\cal D}$.

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 $\Delta_k\in{\cal D}$.
These relations imply that 
$G^{(N)}_{j,j'}(\Delta_k)\to G^{(0)}_{j,j'}(\Delta_k)$
for all $\Delta_k\in{\cal D}$.

\medskip
Let us define for all $N=0,1,2,\dots$ and our regular system
${\cal D}=\{\Delta_k,\;1\le |k|\le M\}$ the Gaussian random
vector
$$
Z_N({\cal D})=\left(\textrm{Re}\,Z_{G^{(N)},j}(\Delta_k),\,
\textrm{Im}\,Z_{G^{(N)},j}(\Delta_k),\quad |k|\le M, \;\; 1\le j\le d\right)
$$
I claim that the elements of the covariance matrices of
the random vectors 
$Z_N({\cal D})$ 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 $Z_N({\cal D})$ converge 
to the covariance matrix of $Z_0({\cal D})$ as $N\to\infty$.

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 ${\cal D}$. Also the properties of the regular systems
imply that if $\Delta_k,\Delta_l\in{\cal D}$, 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 $Z_N({\cal D})$ converge to
the covariance matrix of $Z_0({\cal D})$ as $N\to\infty$.

The convergence of the covariance matrices of the Gaussian random
vectors $Z_N({\cal D})$ with expectation zero also implies that
the distributions of $Z_N({\cal D})$ converge weakly to the
distribution of $Z_0({\cal D})$ as $N\to\infty$. But then the
same can be told about any continuous functions of the coordinates
of the random vectors $Z_N({\cal D})$. Because of the definition
of the Wiener--It\^o integrals of simple functions the random
variables $Y_N$ in formula~(\ref{8.16}) are polynomials, hence continuous
functions of the coordinates of the random vectors $Z_N({\cal D})$.
Besides, these polynomials do not depend on the parameter~$N$. Hence
the previous results imply that formula~(\ref{8.16}) holds. 
Proposition~8.1 is proved.

\medskip
To simplify the application of Proposition~8.1 we also prove 
the following lemma.

\medskip\noindent
{\bf Lemma 8.2.} {\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 8.2 can be considered as the 
multidimensional version of the statement that the limit 
of locally finite measures on ${\mathbb R}^\nu$ with respect to the 
vague convergence is a locally finite measure.

\medskip\noindent
{\it Proof of Lemma 8.2}\/ 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{8.17}
\end{equation}
The identity in~(\ref{8.17}) 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{8.17}) 
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{8.17}) 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 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~8.2 is proved.

\medskip
Proposition~8.1 is a useful tool to prove non-central 
limit theorems for non-linear functionals of vector 
valued Gaussian stationary random fields. In the 
application of this result we have to check some 
properties of the matrix valued spectral measure of 
a Gaussian stationary random field with some nice 
properties. In~\cite{6} we met a similar problem 
about scalar valued Gaussian stationary random fields. 
In that paper we have proved that if the correlation 
function of a scalar valued Gaussian stationary 
random field has some nice properties, then its 
spectral measure satisfies the properties needed in 
the proof of our non-central limit theorem. In the 
generalization of this result to vector valued 
stationary random fields we must get a good control 
on the behaviour of a matrix valued spectral measure 
of a vector valued stationary random field if its 
correlation function has some nice properties. Such 
a result is proved in paper~\cite{13}, where we prove 
the multivariate version of the result in paper~\cite{6}.

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\end{document}



