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%\doi{10.1214/154957805100000186}
%\pubyear{2005}
%\volume{2}
%\firstpage{448}
%\lastpage{505}

\begin{document}
\begin{frontmatter}

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

\title{Non-central limit theorem for non-linear functionals
of vector valued Gaussian stationary random fields}

\runtitle{non-central limit theorem}

\begin{aug}
\author{\fnms{P\'eter} \snm {Major}\corref{}
\ead[label=e1]{major@renyi.hu}}
%\ead[label=e2,url]{www.renyi.hu/$\sim$major}}
\address{Alfr\'ed R\'enyi Mathematical Institute of the Hungarian
Academy of Sciences\\
\printead{e1}\\
url: \href{{http://www.renyi.hu/~major}}{http://www.renyi.hu/$\sim$major}}
%\printead{e2}}
\end{aug}

\runauthor{P\'eter Major}

\begin{abstract} 
Here I prove non-central limit theorems for
non-linear functionals of vector valued stationary random fields
under appropriate conditions. They are the multivariate versions 
of the results in paper~\cite{2}. Previously  A. M. Arcones 
formulated a theorem in paper~\cite{1} which can be considered 
as the multivariate generalization of these results. But I 
found Arcones' discussion incomplete, and in my opinion  to 
give a complete proof first a more profound foundation of the 
theory of vector valued Gaussian stationary random fields has 
to be worked out. This was done in my paper~\cite{4} which 
enabled me to adapt the method in paper~\cite{2} to the study 
of the vector valued case. Here I prove with its help the 
desired multivariate version of the results in paper~\cite{2}. 
\end{abstract}

\begin{keyword}[class=AMS]
\kwd[Primary ] {60F99}
\kwd[; secondary ]{60G60}
\end{keyword}

\begin{keyword}
\kwd{non-central limit theorems}
\kwd{matrix valued spectral measure}
\kwd{vector valued random spectral measure}
\kwd{multiple Wiener--It\^o integrals}
\kwd{shift transformation}
kwd{vague convergence of measures}
\end{keyword}

% history:
%\received{\smonth{1} \syear{2005}}

\end{frontmatter}

\section{Formulation of the results}

Let us consider a $d$-dimensional vector valued
Gaussian stationary random field $X(p)=(X_1(p),\dots,X_d(p))$, 
$p\in{\mathbb Z}^\nu$, where ${\mathbb Z}^\nu$ denotes the 
lattice points with integer coordinates in the $\nu$-dimensional 
Euclidean space ${\mathbb R}^\nu$, and a function $H(x_1,\dots,x_d)$ of $d$ 
variables. We define with their help 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
$$
S_N=A_N^{-1}\sum_{p\in B_N} Y(p) 
$$
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{1.1}
\end{equation}
We prove a non-Gaussian limit theorem for these 
normalized sums $S_N$ with an appropriate norming constant 
$A_N$ if this vector valued Gaussian stationary random field 
$X(p)$, $p\in{\mathbb Z}^\nu$, and function $H(x_1,\dots,x_d)$  
satisfy certain conditions. In~\cite{2} we proved such limit 
theorems for non-linear functionals of scalar valued stationary 
Gaussian random fields, and now we are looking for their natural 
multivariate version.

A. M. Arcones formulated such a result in Theorem~6 of
paper~\cite{1}, but I found his discussion unsatisfactory. 
He applied several notions and results which were not worked 
out. More precisely, there were analogous definitions and 
results in scalar valued models, but in the vector valued 
case their elaboration was missing.  My goal in this paper 
was to give a precise formulation and proof of Arcones' result.

The first step of this program was to work out the theory of
vector valued Gaussian stationary random fields, to present 
the most important notions and results. This was done in my
work~\cite{4}. Here I show how one can 
get the desired limit theorems for non-linear functionals of 
vector valued Gaussian stationary random fields with its help.  
The work~\cite{4} enabled me to adapt the proof 
of~\cite{2} in the study of the multivariate version of the 
result in that paper. 

\medskip\noindent
{\it Remark:}\/ It was professor Herold Dehling who asked 
me to clarify the proof of Theorem~6 in Arcones' paper~\cite{1}. 
The goal of this work together with the preliminary paper~\cite{4} 
was to answer Dehling's question. 
It turned out that to settle this problem first the theory of 
vector valued stationary Gaussian random fields has to be 
worked out. This theory is similar to the theory of the 
scalar valued Gaussian random fields, but there are also 
some essential differences between them. Hence the theory of 
vector valued stationary Gaussian random fields cannot be 
considered as a simple generalization of the theory in the 
scalar valued case. I am grateful to Professor Dehling for 
calling my attention to this problem.  

\medskip
I start the discussion with a short overview of the results 
in~\cite{4} we need in our investigation.

We are working with  $d$-dimensional vector valued Gaussian 
stationary random fields $X(p)=(X_1(p),\dots,X_d(p))$, 
$p\in{\mathbb Z}^\nu$, where ${\mathbb Z}^\nu$ denotes 
the lattice points with integer coordinates in the 
$\nu$-dimensional Euclidean space ${\mathbb R}^\nu$ with 
expectation $EX_j(0)=0$ for all $1\le j\le d$. The distribution 
of such random fields is determined by the covariance function 
$r_{j,j'}(p)=EX_j(0)X_{j'}(p)=EX_j(m)X_{j'}(m+p)$, 
$1\le j,j'\le d$, $m,p\in{\mathbb Z}^\nu$. Our first result
in~\cite{4} was the representation of the functions $r_{j,j'}(p)$
as the Fourier transform $r_{j,j'}(p)=\int e^{i(p,x)}G_{j,j'}(\,dx)$,
of the coordinates of a positive semidefinite matrix valued 
measure $G=(G_{j,j'})$, $1\le j,j'\le d$, on the torus $[-\pi,\pi)^\nu$.
(For a more detailed discussion of this result see Section~2
in~\cite{4}.) Then we defined in Section~3 a vector valued random
spectral measure $Z_G=(Z_{G,1},\dots,Z_{G,d})$ corresponding to a
matrix valued spectral measure $G$ together with a random 
integral with respect to it in such a way that the formula
$X_j(p)=\int e^{i(p,x)}Z_{G,j}(\,dx)$, $p\in{\mathbb Z}^\nu$,
$1\le j\le d$, defines a $d$-dimensional stationary random
field with matrix valued spectral measure~$G$. In Section~4
of~\cite{4} we constructed so-called generalized vector valued 
Gaussian stationary random fields, and proved results analogous 
to the results of Sections~2 and~3 in~\cite{4} for them. In 
particular, we defined the vector valued random spectral measures 
corresponding to the matrix valued spectral measure of a 
generalized vector valued Gaussian stationary random field. 
We need these results, because we define the limit in our 
limit theorems by means of multiple Wiener--It\^o integrals 
with respect to a vector valued random spectral measure 
corresponding to the matrix valued spectral measure of a 
generalized vector valued Gaussian stationary random field. 

Our goal in the subsequent part of work~\cite{4} was to 
give a good representation of those random variables with 
finite second moments which are measurable with respect 
to the $\sigma$-algebra generated by the random variables 
of the underlying vector valued random field together with 
the shift transformation acting on them, because this gives 
a great help in the study of the limit theorems we are 
interested in. With such an aim we defined multiple 
Wiener--It\^o integrals with respect to the coordinates of 
vector valued random spectral measures  in Section~5, and 
studied non-linear functionals of vector valued stationary 
random fields with their help. In Section~6 we proved a
technical result, called the diagram formula, about the 
calculation of the product of two multiple  Wiener--It\^o 
integrals. We needed this result to express Wick polynomials 
of the random variables in our vector valued Gaussian 
stationary random fields. The definition of Wick polynomials, 
which are the natural multivariate generalizations of Hermite 
polynomials, was recalled in Section~7 together with their most 
important properties. Section~7 of~\cite{4} also contains the 
formula about the expression of Wick polynomials by means of
multiple Wiener--It\^o integrals and an important formula
about the calculation of the shift transformations of
random variables given in the form of multiple Wiener--It\^o
integrals. This enabled us to reformulate our limit problems
to a problem about limit theorems for the distribution of a 
sequence of sums of multiple Wiener--It\^o integrals. To 
investigate such problems we proved a result in Section~8 
of~\cite{4} which plays an important role in the investigation 
of this paper. We recalled it in Proposition~2A of this work.

\medskip
First I formulate the conditions I impose on the stationary
field $X(p)$, $p\in{\mathbb Z}^\nu$, and function 
$H(x_1,\dots,x_d)$  I am working with in this paper, and 
formulate the main result. I present the proof in the next 
section. In the proof I shall refer to~\cite{3} instead 
of~\cite{2}, where the results I need in the proof are worked 
out in more detail.

In our results we impose a condition on the behaviour of the
covariance function $r_{j,j'}(p)=EX_j(0)X_{j'}(p)$, 
$1\le j,j'\le d$, $p\in{\mathbb Z}^\nu$, of the vector valued
Gaussian stationary random field $X(p)=(X_1(p),\dots,X_d(p))$, 
$p\in{\mathbb Z}^\nu$. In the corresponding result for 
scalar valued stationary random fields we worked with random
variables of expectation zero, and we imposed a condition
which meant that for large values $p$ the covariance function 
$r(p)=EX(0)X(p)$ behaves like $|p|^{-\alpha}L(p)$ multiplied 
by a function depending on the direction $\frac p{|p|}$ of 
the vector~$p$. Here $L(\cdot)$ is a slowly varying function
at infinity. In the case of vector valued stationary 
random fields we impose the natural multivariate version 
of this condition. Namely, we demand that
$EX_j(p)=0$ for all $1\le j\le d$, $p\in{\mathbb Z}^\nu$, 
and the covariance function 
$r_{j,j'}(p)=EX_j(0)X_{j'}(p)$ satisfies the relation
\begin{equation}
\lim_{T\to\infty}\sup_{p\colon\;p\in{\mathbb Z}^\nu,\,|p|\ge T}
\frac{\left|r_{j,j'}(p)-a_{j,j'}(\frac p{|p|})|p|^{-\alpha}L(|p|)\right|}
{|p|^{-\alpha}L(|p|)}=0 \label{1.2}
\end{equation}
for all 
$1\le j,j'\le d$, where $0<\alpha<\nu$, $L(t)$, $t\ge1$, 
is a real valued function, slowly varying at infinity, 
bounded in all finite intervals, and $a_{j,j'}(t)$ is a 
real valued continuous function on the unit sphere 
${\cal S}_{\nu-1}=\{x\colon\;x\in {\mathbb R}^\nu,\;|x|=1\}$,
and the identity $a_{j',j}(x)=a_{j,j'}(-x)$ holds 
for all $x\in{\cal S}_{\nu-1}$ and $1\le j,j'\le d$. 

\medskip\noindent
{\it Remark.} I show that there are interesting vector
valued Gaussian stationary random fields which satisfy 
condition~(\ref{1.2}). We can construct random fields with 
such a distribution by defining their matrix valued 
spectral measures in the following way. We get a correlation 
function satisfying condition~(\ref{1.2}) with the help of a 
matrix valued spectral measure whose coordinates 
$G_{j.j'}$, $1\le j,j'\le d$, have a density function of the 
form $g_{j,j'}(u)=|u|^{\alpha-\nu}b_{j,j'}(\frac u{|u|})h(u)$ 
with respect to the Lebesgue measure on the torus, 
$u\in [-\pi,\pi)^\nu$,  where $b_{j,j'}(\cdot)$ is a 
non-negative smooth function on the unit sphere 
$\{u\colon\; u\in {\mathbb R}^\nu,\,|u|=1\}$, and $h(u)$ 
is a non-negative, smooth and even function on the torus 
$[-\pi,\pi)^\nu$ which does not  disappear at the origin. 
The functions $b_{j,j'}(\cdot)$ must satisfy some additional
conditions to guarantee that $G=(G_{j,j'})$, $1\le j,j'\le d$,
is a matrix valued spectral measure. We get a good
covariance matrix by defining 
$r_{j,j'}(p)=\int e^{i(p,x)}g_{j,j'}(x)\,dx$, and this 
function satisfies relation~(1.2). Results of this type 
are studied in the theory of generalized functions.  

\medskip
Next we have to formulate a good condition on the function 
$H(x_1,\dots,x_d)$. In scalar valued models first the special
case $H(x)=H_k(x)$ was considered, where $H_k(x)$ denotes the 
$k$-th Hermite polynomial with leading coefficient~1. Then
it was shown that our limit problem with a function $H(x)$ 
whose expansion in the Hermite polynomials has the
form $H(x)=\sum_{l=k}^\infty c_l H_l(x)$ with starting index~$k$
in the summation can be simply reduced to the special case 
when $H(x)=H_k(x)$. We shall prove a similar result in the 
multivariate case. In this case the Wick polynomials take 
the role of the Hermite polynomials. But we shall work 
with Wick polynomials only in an implicit way. We shall 
choose such models where the Wick polynomials can be 
simply calculated. We shall consider such vector valued 
Gaussian stationary random fields 
$X(p)=(X_1(p),\dots,X_d(p))$, $p\in{\mathbb Z}^\nu$, 
whose covariance function satisfies besides condition~(\ref{1.2}) 
also the relation
\begin{equation}
EX_j^2(0)=1 \textrm{ for all } 1\le j\le d, \textrm{ and } 
EX_j(0)X_{j'}(0)=0
\textrm{ if } j\neq j', \;\;1\le j,j'\le d. \label{1.3}
\end{equation}
First I show that this new condition does not mean a real 
restriction of our problem.

\medskip
Let $X(p)=(X_1(p),\dots,X_d(p))$, $p\in{\mathbb Z}^\nu$,
be a vector valued Gaussian stationary random field 
with expectation $EX_j(p)=0$, $p\in{\mathbb Z}^\nu$, 
$1\le j\le d$, and take the random variables 
$X_1(0),\dots,X_d(0)$ in it. We can choose an appropriate 
number $1\le d'\le d$, and $d'$ random variables
$X'_j(0)=\sum_{l=1}^dc_{j,l}X_l(0)$, $1\le j\le d'$, with 
appropriate coefficients $c_{j,l}$, $1\le j\le d'$, 
$1\le l\le d$ in such a way that they have the 
following properties. $EX'_j(0)X'_{j'}(0)=\delta_{j,j'}$, 
$1\le j,j'\le d'$, where $\delta_{j,j'}=0$ if $j\neq j'$, 
and $\delta_{j,j}=1$, and also the random variables 
$X_j(0)$ can be expressed as the linear combination 
of the random variables $X'_l(0)$, $1\le l\le d'$, i.e. 
$X_j(0)=\sum_{l=1}^{d'}d_{j,l}X'_l(0)$ for all 
$1\le j\le d$ with appropriate coefficients $d_{j,l}$.

Let us define the vector valued random field 
$X'(p)=(X'_1(p),\dots,X'_{d'}(p))$ as
$X'_j(p)=\sum_{l=1}^dc_{j,l}X_l(p)$, $1\le j\le d'$, with
the same coefficients~$c_{j,l}$ which appeared in the
definition of $X'_j(0)$ for all $p\in{\mathbb Z}^\nu$. 
Then it is not difficult to see that $X'(p)$, 
$p\in{\mathbb Z}^\nu$, is a $d'$-dimensional Gaussian 
stationary random field whose elements have 
expectation~zero, and it satisfies relation~(\ref{1.3}) 
(with parameter~$d'$ instead of~$d$.) Moreover, if 
the covariance function of the original random field 
$X(p)$ satisfied relation~(\ref{1.2}), then this
new random field also satisfies this condition 
with appropriate new functions 
$a'_{j,j'}(\frac p{|p|})$. Besides, it is not 
difficult to find such a function 
$H'(x_1,\dots,x_{d'})$ for which 
$H'(X'_1(p),\dots,X'_{d'}(p))=H(X_1(p),\dots,X _d(p))$
for all $p\in{\mathbb Z}^\nu$. This means that
with the introduction of this new random field
$X'(p)=(X'_1(p)\dots,X'_{d'}(p))$ we can reformulate
our problem in such a way that our vector valued 
stationary Gaussian random field satisfies both 
relations~(\ref{1.2}) and~(\ref{1.3}). We shall work 
with such a new $d'$-dimensional random field $X'(p)$ 
and function $H'(x_1,\dots,x_{d'})$, only we shall 
omit the sign prime everywhere. 

\medskip
First we consider the case when our function 
$H(x_1,\dots,x_d)$ depends on a previously fixed
constant $k$, and it has the form
\begin{eqnarray}
H(x_1,\dots,x_d)&=&H^{(0)}(x_1,\dots,x_d) \label{1.4} \\
&=&\sum_{\substack{(k_1,\dots, k_d),\; k_j\ge 0,\;1\le j\le d,\\
k_1+\cdots+k_d=k}}
c_{k_1,\dots,k_d}H_{k_1}(x_1)\cdots H_{k_d}(x_d)
\nonumber 
\end{eqnarray}
with some coefficients $c_{k_1,\dots,k_d}$ where $H_k(\cdot)$ 
denotes the $k$-th Hermite polynomial with leading coefficient~1.

In scalar valued models we have proved a non-central limit 
theorem if $H(x)=H_k(x)$, $k\ge2$, and the covariance 
function $r(n)=EX_0X_n$ satisfies condition~(\ref{1.2}) with 
$0<\alpha<\frac\nu k$ in the special case $d=1$. This result 
was formulated in Theorem~8.2 of~\cite{3}. In this result 
the limit was described with the help of a $k$-fold 
Wiener--It\^o integral with respect to an appropriate 
random spectral measure. This random spectral measure 
corresponds to that spectral measure which appeared in 
Lemma~8.1 of~\cite{3} as the limit of a sequence consisting 
of appropriately normalized versions of the spectral measure 
of a stationary random field $X(p)$, $p\in{\mathbb Z}^\nu$, 
which satisfies condition~(\ref{1.2})in the case $d=1$. 
I shall prove a multivariate version of Theorem~8.2 
of~\cite{3}. But to do this first I present a multivariate 
version of Lemma~8.1 in~\cite{3}. 

This generalization of Lemma~8.1 in~\cite{3} describes 
the limit behaviour  of the matrix valued spectral measure 
of a vector valued random field whose covariance function 
satisfies formula~(\ref{1.2}). This limit behaviour is 
described with the help of an appropriate spectral measure 
which is the spectral measure of a vector valued generalized
stationary random field. This new spectral measure and a 
random spectral measure corresponding to it appears in 
the definition of th Wiener--It\^o integrals that describe 
the limit in the multivariate version of the limit theorem 
in Theorem~8.2 of~\cite{3}.

\medskip
Given a vector valued stationary random field 
$X(p)=(X_1(p),\dots,X_d(p))$, $p\in{\mathbb Z}^\nu$, 
with expectation zero and covariance function 
$r_{j,j'}(p)=EX_j(0)X_{j'}(p)$ that satisfies relation~(\ref{1.2})
let us consider its matrix valued spectral measure $(G_{j,j'})$,
$1\le j,j'\le d$, on the torus $[-\pi,\pi)^\nu$. (The existence 
of this matrix valued spectral measure, and its 
characterization was proved e.g. in Theorem~2.2 of~\cite{4}.) 
Take its rescaled version
\begin{equation}
G^{(N)}_{j,j'}(A)=\frac{N^\alpha}{L(N)}G_{j,j'}\left(\frac AN\right),
\quad A\in{\cal B}^\nu, \quad N=1,2,\dots,\;\;1\le j,j'\le d, \label{1.5}
\end{equation}
concentrated on $[-N\pi,N\pi)^\nu$ for all $N=1,2,\dots$, 
where ${\cal B}^\nu$ denotes the $\sigma$-algebra of the 
Borel measurable sets on ${\mathbb R}^\nu$. In the next 
Proposition I formulate a result which states that these 
complex measures $G^{(N)}_{j,j'}$ have a vague limit. This 
result also describes some properties of this limit. 
Before formulating it I recall the definition of vague limit.

\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 if
$$
\lim_{N\to\infty}\int f(x)\,G^{(N)}(\,dx)=\int f(x)\,G^{(0)}(\,dx) 
$$
for all continuous functions~$f$ on ${\mathbb R}^\nu$ with a 
bounded support.}

\medskip
The definition of complex measures with locally finite
total variation together with the notion of vector valued 
Gaussian stationary generalized random fields and their 
matrix valued spectral measures was introduced in Section~4 
of~\cite{4}. In the next result I formulate the multivariate 
version of Lemma~8.1 in~\cite{3} with their help.

\medskip\noindent
{\bf Proposition 1.1.} {\it Let $G=(G_{j,j'})$ be the matrix
valued spectral measure of a $d$-dimensional vector valued 
stationary random field whose covariance function 
$r_{j,j'}(p)$  satisfies relation~(\ref{1.2}) with some parameter
$0<\alpha<\nu$. Then for all pairs $1\le j,j'\le d$
the sequence of complex measures~$G^{(N)}_{j,j'}$ 
defined in~(\ref{1.5}) with the help of the complex measure 
$G_{j,j'}$ tends vaguely to a complex measure~$G^{(0)}_{j,j'}$ 
on ${\mathbb R}^\nu$ with locally finite total variation. These 
complex measures~$G^{(0)}_{j,j'}$, $1\le j,j'\le d$, 
have the homogeneity property
\begin{equation}
G^{(0)}_{j,j'}(A)=t^{-\alpha}G^{(0)}_{j,j'}(tA) 
\quad \textrm{for all bounded } A\in{\cal B}^\nu, 
\;1\le j,j'\le d,\textrm{ and } t>0.  \label{1.6}
\end{equation}
The complex measure $G^{(0)}_{j,j'}$ with locally 
finite variation is determined by the number $0<\alpha<\nu$
and the function $a_{j,j'}(\cdot)$ on the unit sphere~$S_{\nu-1}$ 
introduced in formula~(\ref{1.2}).

There exists  a vector valued Gaussian stationary generalized
random field on ${\mathbb R}^\nu$ with that matrix valued spectral 
measure  $(G^{(0)}_{j,j'})$, $1\le j,j'\le d$, whose coordinates 
are the above defined complex measures $G^{(0)}_{j,j'}$, 
$1\le j,j'\le d$.} 

\medskip\noindent
{\it Remark.}\/ The statement of Proposition~1.1 about the measures
$G_{j,j}$ is actually contained in Lemma~8.1 of~\cite{3}. On the other
hand, Proposition~1.1 states a similar result on the rescaled
versions of the complex measures $G_{j,j'}$ also in the case
$j\neq j'$. The main problem in the proof of this statement is that
in this case $G_{j,j'}$ need not be a (real valued, positive) measure. 
On the other hand, it is the element of a matrix valued positive 
definite (spectral) measure, and this fact plays an important 
role in the proof of Proposition~1.1.

\medskip
It was proved in~\cite{4} that to each matrix 
valued spectral measure corresponds a vector valued random 
spectral measure such that the Fourier transform of this
random spectral measure defines a Gaussian stationary random
field whose spectral measure equals this spectral measure. 
We can define  multiple Wiener--It\^o integrals with respect 
to the coordinates of this vector valued random spectral 
measure. This enables to formulate the multivariate version 
of Theorem~8.2 in~\cite{3} in the following Theorem~1.2A. 
This is a limit theorem where the limit is defined by means 
of a sum of multiple Wiener--It\^o integrals with respect to 
the coordinates of a vector valued random spectral measure 
$Z_{G^{(0)}}=(Z_{G^{(0)},1},\dots,Z_{G^{(0)},d})$ which corresponds 
to the matrix valued spectral measure $(G^{(0)}_{j,j'})$, 
$1\le j,j'\le d$, defined in Proposition~1.1.

\medskip\noindent
{\bf Theorem 1.2A.} {\it Fix some integer $k\ge1$,
and let $X(p)=(X_1(p),\dots,X_d(p))$, 
$p\in{\mathbb Z}^\nu$, be a vector valued Gaussian 
stationary random field whose covariance function 
$r_{j,j'}(p)=EX_j(0)X_{j'}(p)$, $1\le j,j'\le d$, 
$p\in{\mathbb Z}^\nu$, satisfies relation~(\ref{1.2}) with 
some $0<\alpha<\frac\nu k$ and relation~(\ref{1.3}). Let 
$H(x_1,\dots,x_d)$ be a function of the form given 
in~(\ref{1.4}) with the parameter $k$ we have fixed in the
formulation of this result. Define the random variables 
$Y(p)=H(X_1(p),\dots,X_d(p))$ for all 
$p\in{\mathbb Z}^\nu$ together with their normalized 
partial sums 
$$
S_N=\frac1{N^{\nu-k\alpha/2}L(N)^{k/2}}\sum_{p\in B_N}Y(p),
$$ 
where the set $B_N$ was defined in~(\ref{1.1}). These random 
variables $S_N$, $N=1,2,\dots$, satisfy the following 
limit theorem.

Let $Z_{G^{(0)}}=(Z_{G^{(0)},1},\dots,Z_{G^{(0)},d})$ be a vector 
valued random spectral measure which corresponds to the 
matrix valued spectral measure $(G^{(0)}_{j,j'})$, 
$1\le j,j'\le d$, defined in Proposition~1.1 with
the help of the matrix valued spectral measure $G=(G_{j,j'})$,
corresponding the covariance function $r_{j,j'}(p)$ we
are working with. Then the sum of multiple Wiener--It\^o 
integrals
\begin{eqnarray}
S_0&=&\sum_{\substack{(k_1,\dots, k_d),\; k_j\ge 0,\;1\le j\le d, \\
k_1+\cdots+k_d=k}} c_{k_1,\dots,k_d}
\int \prod_{l=1}^\nu\frac{e^{i(x_1^{(l)}+\cdots+x_k^{(l)})}-1}
{i(x^{(l)}_1+\cdots+x_k^{(l)})} \label{1.7} \\
&&\qquad\qquad\qquad Z_{G^{(0)},{j(1|k_1,\dots,k_d)}}(\,dx_1)\dots 
Z_{G^{(0)},{j(k|k_1,\dots,k_d)}}(\,dx_k)
\nonumber
\end{eqnarray}
exists, where we use the notation  
$x_p=(x_p^{(1)},\dots,x_p^{(\nu)})$, $p=1,\dots,k$, 
and we define the indices $j(s|k_1,\dots,k_d)$,
$1\le s\le k$, as  $j(s|k_1,\dots,k_d)=r$ if 
$\sum_{u=1}^{s-1} k_u< r\le \sum_{u=1}^s k_u$, 
$1\le s\le k$. (For $s=1$ we apply the notation 
$\sum_{u=1}^{0} k_u=0$ in the definition of 
$j(1|k_1,\dots,k_d)$.) The normalized sums 
$S_N$ converge in distribution to the random 
variable $S_0$ defined in~(\ref{1.7}) as $N\to\infty$.}

\medskip
I explain the indexation of the terms
$Z_{G^{(0)},{j(s|k_1,\dots,k_d)}}(\,dx_s)$ in formula (\ref{1.7}) 
in a simpler way. In the first $k_1$ variables $x_1,\dots,x_{k_1}$ 
we wrote $Z_{G^{(0)},1}(\,dx_s)$, $1\le s\le k_1$, in the next $k_2$ 
terms we wrote $Z_{G^{(0)},2}(\,dx_s)$, $k_1+1\le s\le k_1+k_2$, and 
so on. In the last $k_d$ terms we wrote $Z_{G^{(0)},d}(\,dx_s)$,
$k_1+\cdots+k_{d-1}+1\le s\le k$.


\medskip
In Theorem 1.2A we described the limit of 
$A_N^{-1}\sum_{p\in B_N} H(X_1(p),\dots,X_d(p))$ if the 
expansion of the functions $H(x_1,\dots,x_d)$ in
the product of Hermite polynomials contains only polynomials 
of order~$k$. The next Theorem~1.2B which is the multivariate
version of Theorem~8.2$'$ in~\cite{3} states that a 
result similar to Theorem~1.2A holds if the expansion of 
the function $H(x_1,\dots,x_d)$ in the product of Hermite 
polynomials may also contain polynomials of higher order.

\medskip\noindent
{\bf Theorem 1.2B.} {\it Let us consider the same vector 
valued Gaussian stationary random field 
$X(p)=(X_1(p),\dots,X_d(p))$, $p\in{\mathbb Z}^\nu$, as
in Theorem~1.2A together with a function of the form 
$H(x_1,\dots,x_d)=H^{(0)}(x_1,\dots,x_d)+H^{(1)}(x_1,\dots,x_d)$,
where $H^{(0)}(x_1,\dots,x_d)$ was defined in~(\ref{1.4}), and
\begin{equation}
H^{(1)}(x_1,\dots,x_d)
=\sum_{\substack{(k_1,\dots, k_d),\; k_j\ge 0,\;1\le j\le d,\\
k_1+\cdots+k_d\ge k+1}} 
c_{k_1,\dots,k_d}H_{k_1}(x_1)\cdots H_{k_d}(x_d) \label{1.8}
\end{equation}
with coefficients $c_{k_1,\dots,k_d}$ such that
\begin{equation}
\sum_{\substack{(k_1,\dots, k_d),\; k_j\ge 0,\;1\le j\le d,\\
k_1+\cdots+k_d\ge k+1}} 
\frac{c^2_{k_1,\dots,k_d}}{k_1!\cdots k_d!}<\infty. \label{1.9}
\end{equation}
Define the random variables 
$Y(p)=H(X_1(p),\dots,X_d(p))$ for all $p\in{\mathbb Z}^\nu$
and their normalized partial sums 
$$
S_N=\frac1{N^{\nu-k\alpha/2}L(N)^{k/2}}\sum_{p\in B_N}Y(p),\quad
N=1,2,\dots,
$$ 
with this function $H(x_1,\dots,x_d)$. The 
random variables $S_N$ converge in distribution to the 
random variable $S_0$ defined in formula~(\ref{1.7}) as $N\to\infty$.}

\medskip
Actually condition~(\ref{1.9}) in Theorem~1.2B means that 
$$
E{H^{(1)}}^2(X_1(0),\dots,X_d(0))<\infty.
$$ 
Finally I mention that Arcones formulated a more general 
result. To present it, more precisely its generalization 
when we are working with stationary random fields 
parametrized by the lattice points of ${\mathbb Z}^\nu$ 
with $\nu\ge1$, let us define the following parameter 
sets for all $N=1,2,\dots$ and 
$t=(t_1,\dots,t_\nu)$, $0\le t_l\le 1$, for all $1\le l\le \nu$.
\begin{equation}
B_N(t)=B_N(t_1,\dots,t_\nu)
=\{p=(p_1,\dots,p_\nu)\colon\; 0\le p_l<Nt_l \textrm { for all } 
1\le l\le\nu\}. \label{1.10}
\end{equation}
With this notation I formulate the following result.

\medskip\noindent
{\bf Theorem 1.3.} {\it Let us consider the same vector 
valued Gaussian stationary random field 
$X(p)=(X_1(p),\dots,X_d(p))$, $p\in{\mathbb Z}^\nu$, 
and function $H(x_1,\dots,x_d)$ as in Theorem~1.2B.
Define the random variables 
$Y_p=H(X_1(p),\dots,X_d(p))$ for all $p\in{\mathbb Z}^\nu$
and the random fields 
\begin{equation}
S_N(t)=\frac1{N^{\nu-k\alpha/2}L(N)^{k/2}}\sum_{p\in B_N(t)}Y(p) \label{1.11}
\end{equation}
with parameter set $t=(t_1,\dots,t_\nu)$, $0<t_l\le 1$,
$1\le l\le\nu$, for all $N=1,2,\dots$, where the set 
$B_N(t)$ was defined in~(\ref{1.10}). The finite dimensional 
distributions of the random fields $S_N(t)$ converge to 
that of the random field $S_0(t)$, $t=(t_1,\dots,t_\nu)$, 
$0\le t_l\le 1$, $1\le l\le\nu$, defined by the formula
\begin{eqnarray}
S_0(t)&=&\sum_{\substack{(k_1,\dots, k_d),\; k_j\ge 0,\;1\le j\le d,\\
k_1+\cdots+k_d=k}}
c_{k_1,\dots,k_d}
\int \prod_{l=1}^\nu\frac{e^{it_l(x_1^{(l)}+\cdots+x_k^{(l)})}-1}
{i(x^{(l)}_1+\cdots+x_k^{(l)})} \label{1.12} \\
&&\qquad\qquad Z_{G^{(0)},{j(1|k_1,\dots,k_d)}}(\,dx_1)\dots 
Z_{G^{(0)},{j(k|k_1,\dots,k_d)}}(\,dx_k)  \nonumber
\end{eqnarray}
if the limit $N\to\infty$ is taken. Similarly to Theorem~1.2A 
we use the notation $x_p=(x_p^{(1)},\dots,x_p^{(\nu)})$, 
$p=1,\dots,k$, and  the indices $j(s|k_1,\dots,k_d)$,
$1\le s\le k$, are defined as in formula~(\ref{1.7}).}  

\medskip
Let us observe that the kernel function in the Wiener--It\^o 
integrals expressing $S_0(t)$ equals $\varphi_t(x_1+\cdots+x_k)$,
where $\varphi_t(u)$, $u\in {\mathbb R}^\nu$, is the Fourier transform 
of the Lebesgue measure on the rectangle 
$[0,t_1]\times\cdots\times[0 ,t_\nu]$. The integral in~(\ref{1.12}) 
is taken on the whole space. (The formulation of Arcones' 
result at this point is erroneous.)

\medskip
We have formulated Theorem~1.3 in the form as Arcones did, but
actually we could have formulated it in a slightly more general way.
We could have defined the sets $B_{N}(t)$ in~(\ref{1.10}), the random
variables $S_N(t)$, $N=1,2,\dots$, in~(\ref{1.11}) and $S_0(t)$ 
in~(\ref{1.12}) for all $t=(t_1,\dots,t_\nu)\in[0,\infty)^\nu$ and 
not only for $t=(t_1,\dots,t_\nu)\in[0,1]^\nu$. We could have proved,
similarly to the proof of Theorem~1.3, that the finite dimensional
distributions of the random fields $S_N(t)$ converge to the
finite dimensional distributions of the random field $S_0(t)$ as
$N\to\infty$ also in this more general case. We can  also say that 
the random field $S_0(t)$, $t\in[0,\infty)^\nu$, is self-similar with 
parameter $\nu-k\alpha/2$, i.e 
$S_0(ut)\stackrel{\Delta}{=}u^{\nu-k\alpha/2}S_0(t)$
for all $u>0$, where $\stackrel{\Delta}{=}$ means that the finite
dimensional distributions of the two random fields agree.

One can prove the self-similarity property of the random field $S_0(t)$,
$t\in[0,\infty)^\nu$, by exploiting that $G^{(0)}(uA)=u^\alpha G^{(0)}(A)$
for the spectral measure $G^{(0)}$ for all $u>0$ and measurable sets
$A\subset{\mathbb R}^\nu$ by formula~(1.6) in Proposition~1.1,
and this implies that 
$$
(Z_{G^{(0)},1}(uA_1),\dots,Z_{G^{(0)},d}(uA_d))\stackrel{\Delta}{=}
(u^{\alpha/2}Z_{G^{(0)},1}(A_1),\dots,u^{\alpha/2}Z_{G^{(0)},d}(A_d))
$$
for all $u>0$ and measurable sets 
$A_1\in{\mathbb R}^\nu,\dots,A_d\in{\mathbb R}^\nu$.
We still have to exploit that the kernel functions
$$
f_t(x_1,\dots,x_d)=\prod_{l=1}^\nu\frac{e^{it_l(x_1^{(l)}+\cdots+x_k^{(l)})}-1}
{i(x^{(l)}_1+\cdots+x_k^{(l)})}
$$ 
in the Wiener--It\^o integrals in~(1.12) (with the notation 
$t=(t_1,\dots,t_\nu)$) have the property 
$$
f_{ut}(x_1,\dots,x_k)=u^\nu f_t(ux_1,\dots,ux_k)
$$ 
for all $u>0$, $t\in[0,\infty)^\nu$, $x_j\in{\mathbb R}^\nu$, 
$1\le j\le k$. The self-similarity
property of the random field $S_0(t)$, $t\in[0,\infty)^\nu$, 
can be proved with the help of the above observations.

\section{Proof of the results}

Let us consider the proof of Theorem~1.2A. A most 
important point in it is to find a good representation of 
the normalized random sums $S_N$ appearing in this result.

Let $X(p)=(X_1(p),\dots,X_d(p))$, 
$p\in{\mathbb Z}^\nu$, be a vector valued Gaussian 
stationary random field whose covariance function 
$r_{j,j'}(p)=EX_j(0)X_{j'}(p)$, $1\le j,j'\le d$, 
$p\in{\mathbb Z}^\nu$, satisfies relation~(\ref{1.2}) with 
some parameter $0<\alpha<\frac\nu k$ and relation~(\ref{1.3}).
Let $G=(G_{j,j'})$, $1\le j,j'\le d$, be the matrix
valued spectral measure of this stationary random
field, and let us consider that vector valued
random spectral measure $Z_G=(Z_{G,1},\dots,Z_{G,d})$
corresponding to this spectral measure for which
$X_j(p)=\int e^{i(p,x)}Z_{G,j}(\,dx)$ for all 
$p\in{\mathbb Z}^\nu$ and $1\le j\le d$.

Because of relation (\ref{1.3}) the random variable
$Y(0)=H(X_1(0),\dots,X_d(0))$ defined with the
function~$H(x_1,\dots,x_d)=H^{(0)}(x_1,\dots,x_d)$ 
introduced in~(\ref{1.4}) is a Wick polynomial of 
order~$k$ of the random variables 
$X_1(0),\dots,X_d(0)$, (see Section~7 in~\cite{4}). 
This Wick polynomial can be rewritten because 
of the multivariate version of It\^o's formula 
(see Corollary of Theorem~7.2 in~\cite{4}) and the identity
$X_j(0)=\int {\mathbb I}_1(y)Z_{G,j}(\,dy)$, $1\le j\le d$, 
where ${\mathbb I}_1(y)$ denotes the indicator function of 
the torus $[-\pi,\pi)^\nu$, as a sum of $k$-fold
multiple Wiener--It\^o integrals with respect
to the vector valued random spectral measure
$Z_G=(Z_{G,1},\dots,Z_{G,d})$. Let us remark that by 
Lemma~8B of~\cite{3} condition~(\ref{1.2}) implies that the
diagonal elements $G_{j,j}$, $1\le j\le d$, of the
matrix valued spectral measure $G=(G_{j,j'})$,
$1\le j,j'\le d$, are non-atomic. Hence we can 
define the multiple Wiener--It\^o integrals with 
respect to the coordinates of the vector valued 
random spectral measure $Z_G=(Z_{G,1},\dots,Z_{G,d})$. 
The above considerations yield the identity
\begin{eqnarray*}
Y_0&=&H(X_1(0),\dots,X_d(0))=H^{(0)}(X_1(0),\dots,X_d(0)) \\
&=&\sum_{\substack{(k_1,\dots, k_d),\; k_j\ge 0,\;1\le j\le d,\\
k_1+\cdots+k_d=k}} 
\colon c_{k_1,\dots,k_d} X_1(0)^{k_1}\cdots X_d(0)^{k_d} \colon \\
&=&\sum_{\substack{(k_1,\dots,k_d),\; k_j\ge0\;1\le j\le d,\\ 
k_1+\cdots+k_d=k}}
c_{k_1,\dots,k_d}\int {\mathbb I}_1(y_1)\dots {\mathbb I}_1(y_k) \\
&& \qquad\qquad\qquad\qquad \prod_{j=1}^d
\left(\prod_{t=k_1+\cdots+k_{j-1}+1}^{k_1+\cdots+k_j} Z_{G,j}(\,dy_t)\right),
\end{eqnarray*}
where for $j=1$ we define
$\prod\limits_{t=k_1+\cdots+k_{j-1}+1}^{k_1+\cdots+k_j} Z_{G,j}(\,dy_t)
=\prod\limits_{t=1}^{k_1} Z_{G,1}(\,dy_t)$, and if $k_j=0$ for some 
$1\le j\le d$, then we drop the term 
$\prod\limits_{t=k_1+\cdots+k_{j-1}+1}^{k_1+\cdots+k_j}Z_{G,j}(\,dy_t)$
from this expression. (Here 
$\colon\! P(X_1(0),\dots,X_d(0))\colon$ denotes the
Wick polynomial corresponding to $P(X_1(0),\dots,X_d(0))$, 
where $P(x_1,\dots,x_d)$ is a homogeneous polynomial.)

Since $Y(p)=T_pY(0)$ with the shift transformation $T_p$ for all 
$p\in{\mathbb Z}^\nu$, the previous
identity and Proposition~7.4 in~\cite{4} yield the formula
\begin{eqnarray*}
Y(p)&=&T_pY(0) \\
&=& \!\!\!\!\!\!\!\!\!\!\!\!\!\!
\sum_{\substack{(k_1,\dots,k_d),\;k_j\ge0,\;1\le j\le d,\\ 
k_1+\cdots+k_d=k}}  \!\!\!\!\!\!\!\!\!\!\!\!\!
 c_{k_1,\dots,k_d}\int e^{i(p,y_1+\cdots+y_k)}\prod_{j=1}^d
\left(\prod_{t=k_1+\cdots+k_{j-1}+1}^{k_1+\cdots+k_j} \!\!\! 
Z_{G,j}(\,dy_t)\right)
\end{eqnarray*}
for all $p\in{\mathbb Z}^\nu$. We get by summing up 
this formula for all $p\in B_N$ that
\begin{eqnarray*}
&&\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!
S_N=\frac1{N^{\nu-k\alpha/2}L(N)^{k/2}}
\sum_{\substack{(k_1,\dots,k_d),\;k_j\ge0,\;1\le j\le d\\ k_1+\cdots +k_d=k}}
 c_{k_1,\dots,k_d} \\
&& \!\!\!\!\!\!\!\!\!\!
\int\prod _{l=1}^\nu \frac{e^{i(N(y_1^{(l)}+\cdots+y_k^{(l)})}-1} 
{e^{i((y_1^{(l)}+\cdots+y_k^{(l)})}-1} \prod_{j=1}^d 
\left(\prod_{t=k_1+\cdots+k_{j-1}+1}^{k_1+\cdots+k_j} Z_{G,j}(\,dy_t)\right),
\end{eqnarray*}
where we write $y=(y^{(1)},\dots,y^{(\nu)})$ for all 
$y\in[-\pi,\pi)^\nu$.  

We rewrite the above sum of Wiener--It\^o integrals with the 
change of variables $x_s=Ny_s$, $1\le s\le k$, in the 
form
\begin{eqnarray}
S_N&=& \sum_{\substack{(k_1,\dots,k_d),\; k_j\ge0,\;1\le j\le d,\\ 
k_1+\cdots +k_d=k}} \int f^N_{k_1,\dots,k_d}(x_1,\dots,x_k)
\nonumber \\
&&\qquad\qquad\qquad\qquad \prod_{j=1}^d 
\left(\prod_{t=k_1+\cdots+k_{j-1}+1}^{k_1+\cdots+k_j} Z_{G^{(N)},j}(\,dx_t)\right),
\label{2.1}
\end{eqnarray}
where
\begin{equation}
f^N_{k_1,\dots,k_d}(x_1,\dots,x_k)=c_{k_1,\dots,k_d}\prod_{l=1}^\nu 
\frac{e^{i((x_1^{(l)}+\cdots+x_k^{(l)})}-1} 
{N(e^{i((x_1^{(l)}+\cdots+x_k^{(l)})/N}-1)} \label{2.2} 
\end{equation}
is a function on $[-N\pi,N\pi)^\nu$, and 
$Z_{G^{(N)},j}(A)=\frac{N^{\alpha/2}}{L(N)^{1/2}}Z_{G,j}(\frac AN)$ 
for all measurable sets $A\subset [-N\pi,N\pi)^\nu$ 
and $j=1,\dots,d$. Here we use the notation 
$x_s=(x_s^{(1)},\dots,x_s^{(\nu)})$ for all $1\le s\le k$
for a set of vectors $(x_1,\dots,x_k)$, $x_s\in {\mathbb R}^\nu$ 
for all $1\le s\le k$.  Let us observe that 
$(Z_{G^{(N)},1},\dots,Z_{G^{(N)},d})$ is a vector valued 
random spectral measure on the torus $[-N\pi,N\pi)^\nu$ 
corresponding to the matrix valued spectral measure 
$G^{(N)}=(G^{(N)}_{j,j'})$, $1\le j,j'\le d$, on the 
torus $[-N\pi,N\pi)^\nu)$, defined by the formula 
$G^{(N)}_{j,j'}(A)=\frac{N^\alpha}{L(N)}G_{j,j'}(\frac AN)$, 
$1\le j,j'\le d$, 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 
$X(p)=(X_1(p),\dots,X_d(p))$, $p\in{\mathbb Z}^\nu$.

In formulas (\ref{2.1}) and (\ref{2.2}) we gave a useful 
representation of the normalized random sum $S_N$ 
investigated in Theorem~1.2A in the form of a sum of 
$k$-fold multiple Wiener--It\^o integrals. Let us 
observe that the kernel functions 
$f^N_{k_1,\dots,k_d}(x_1,\dots,x_k)$ of these 
Wiener--It\^o integrals satisfy the relation
\begin{equation}
\lim_{N\to\infty} f^N_{k_1,\dots,k_d}(x_1,\dots,x_k)
=f^0_{k_1,\dots,k_d}(x_1,\dots,x_k) \label{2.3}
\end{equation}
for all indices $k_1,\dots, k_d$ such that $k_j\ge0$, 
$1\le j\le d$, and $k_1+\cdots+k_d=k$ with the function
\begin{equation}
f^0_{k_1,\dots,k_d}(x_1,\dots,x_k)=c_{k_1,\dots,k_d}\prod_{l=1}^\nu 
\frac{e^{i((x_1^{(l)}+\cdots+x_k^{(l)})}-1} 
{i(x_1^{(l)}+\cdots+x_k^{(l)})} \label{2.4}
\end{equation}
defined on ${\mathbb R}^{k\nu}$, and this convergence is 
uniform in all bounded subsets of ${\mathbb R}^{k\nu}$.

On the other hand, Proposition~1.1 states that the matrix
valued spectral measures $G^{(N)}=(G^{(N)}_{j,j'})$  converge to
a matrix valued spectral measure $G^{(0)}=(G^{(0)}_{j,j'})$ on 
${\mathbb R}^\nu$, and in (\ref{2.1}) we integrate with respect 
to a vector valued random spectral measure corresponding to 
the matrix spectral measure $(G^{(N)}_{j,j'})$, $1\le j,j'\le N$. 
Hence it is natural to expect that the random variables $S_N$ 
converge in distribution to the random variable
\begin{eqnarray}
S_0&=&\sum_{\substack{(k_1,\dots,k_d),\; k_j\ge0,\;1\le j\le d,\\ 
k_1+\cdots +k_d=k}} 
\int f^0_{k_1,\dots,k_d}(x_1,\dots,x_k) \nonumber \\
&&\qquad\qquad\qquad\qquad \prod _{j=1}^d 
\left(\prod_{t=k_1+\cdots+k_{j-1}+1}^{k_1+\cdots+k_j}
Z_{G^{(0)},j}(\,dx_t)\right), \label{2.5}
\end{eqnarray}
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 $(G^{(0)}_{j,j'})$, $1\le j,j'\le d$. This 
is actually the statement of Theorem~1.2A with a slightly 
different notation.

\medskip
First I prove Proposition~1.1, and then Theorem~1.2A by 
justifying the above heuristic argument with the help of 
Proposition~2A presented later, which is a reformulation 
of Proposition~8.1 in~\cite{4}.

\medskip\noindent
{\it Proof of Proposition 1.1.} Let us remark that for the 
diagonal elements $G_{j,j}$, $1\le j\le d$, of the matrix 
valued spectral measure $G$ the measures $G^{(N)}_{j,j}$ 
converge vaguely to a locally finite measure 
$G^{(0)}_{j,j}$ which satisfies relation~(\ref{1.6}). This 
follows from Lemma~8.1 of~\cite{3}. Simply we have to 
apply this lemma for the measures $G_{j,j}$.

In the case of the non-diagonal elements $G_{j,j'}$, 
$j\neq j'$, this argument does not work in itself, because
a complex measure $G_{j,j'}$ with finite total
variation may be not a measure. In this case we exploit
that $G$ is a positive semidefinite matrix valued measure, 
hence the $2\times2$ matrix
$$
G(A|j,j')=\left( \begin{array}{l}
G_{j,j}(A),\quad\,\, G_{j,j'}(A) \\
G_{j',j}(A),\quad G_{j',j'}(A)
\end{array} \right)
$$
is positive semidefinite for all pairs $1\le j,j'\le d$, $j\neq j'$,
and measurable sets $A\subset {\mathbb R}^\nu$. Hence the 
quadratic forms
$(1,1)G(A|j,j')(1,1)^*=G_{j,j}(A)+G_{j',j'}(A)+G_{j,j'}(A)+G_{j',j}(A)$
and
$(1,i)G(A|j,j')(1,-i)^*=G_{j,j}(A)+G_{j',j'}(A)-i[G_{j,j'}(A)-G_{j',j}(A)]$
are non-negative numbers for all measurable sets 
$A\subset {\mathbb R}^{\nu}$. This implies that the set-functions 
$R_{j,j'}(\cdot)$ and $S_{j,j'}(\cdot)$ defined as 
$R_{j,j'}(A)=G_{j,j}(A)+G_{j',j'}(A)+G_{j,j'}(A)+G_{j',j}(A)$ and
$S_{j,j'}(A)=G_{j,j}(A)+G_{j',j'}(A)-i[G_{j,j'}(A)-G_{j',j}(A)]$ for 
all measurable sets~$A$ are finite measures. Moreover, they 
satisfy relation~(\ref{1.2}) (with a new function $a_{j,j'}(\cdot)$). 
Hence Lemma~8.2 of~\cite{3} can be applied for them. This fact 
together with the property
$G_{j',j}(A)=\overline{G_{j,j'}(A)}$ and the result about the
behaviour of $G^{(N)}_{j,j}$ and $G^{(N)}_{j',j'}$ imply that 
the sequence $G_{j,j'}^{(N)}$ vaguely converges to a complex 
measure $G^{(0)}_{j,j'}$ with locally finite total 
variation which satisfies relation~(\ref{1.6}). (A similar 
argument was applied in the proof of Proposition~8.1 in~\cite{4}.)

Lemma~8.1 in~\cite{3} (applied for the measures 
$G_{j,j}$, $1\le j\le d$) implies that 
$\mu^{(0)}_{j,j}(\,dx)=\prod_{l=1}^\nu\frac{1-\cos x^{(l)}}{(x^{(l)})^2}
\,G^{0)}_{j,j}(\,dx)$ is a finite measure whose Fourier 
transform can be expressed by means of the parameter $\alpha$ 
and the function $a_{j,j}(x)$ on the unit sphere~${\cal S}_{\nu-1}$. 
This fact together with relation~(\ref{1.6}) imply that 
these quantities determine the distribution of the measure 
$G^{(0)}_{j,j}$. By refining this argument one can prove that 
$\mu^{(0)}_{j,j'}(\,dx)=\prod_{l=1}^\nu\frac{1-\cos x^{(l)}}{(x^{(l)})^2}
\,G^{0)}_{j,j'}(\,dx)$ is a complex measure with 
finite total variation whose Fourier transform is determined
by the parameter $\alpha$ and the function $a_{j,j'}(\cdot)$ 
on the unit sphere ${\cal S}_{\nu-1}$. Hence they determine 
the distribution of the complex measure 
$G^{(0)}_{j,j'}$. In a detailed proof the Fourier transform of 
the complex measures $\mu^{(0)}_{j,j'}$ should be
written down. In Lemma~8.1 of~\cite{3} this is done for the 
corresponding formula. Here I omitted this part of the proof, 
because actually this result is not needed in our 
investigation.

We have to show that $(G^{(0)}_{j,j'})$, $1\le j,j'\le d$
is a matrix valued spectral measure on ${\mathbb R}^\nu$. (In 
Section~4 of~\cite{4} the matrix valued spectral measures on 
${\mathbb R}^\nu$ are defined as the positive semidefinite matrix
valued even measures on ${\mathbb R}^\nu$ with moderately 
increasing distribution at infinity.) In Lemma~8.2 of~\cite{4} 
it is proved that they are positive semidefinite matrix valued 
even measures on ${\mathbb R}^\nu$, since their coordinates 
$G^{(0)}_{j,j'}$ are the vague limits of the coordinates 
$G^{(N)}_{j,j'}$ of the matrix valued spectral measures 
$(G^{(N)}_{j,j'})$, $1\le j,j'\le d$ as $N\to\infty$. It 
follows from relation~(\ref{1.6}) that they have moderately 
increasing  distribution at infinity. (This property is 
defined in formula~(4.1) in~\cite{4}.) Proposition~1.1 is proved.

\medskip
To prove Theorem 1.2A I recall Proposition~8.1 of~\cite{4}
in the following Proposition~2A. It states that under
certain conditions a sequence of sums of $k$-fold
Wiener--it\^o integrals converges in distribution to
a sum of $k$-fold Wiener--It\^o integrals. 
Before formulating this result I recall that in 
Section~5 of~\cite{4}, in the definition of multiple 
Wiener--It\^o integrals I have defined a real 
Hilbert space ${\cal K}_{k,j_1,\dots,j_k}
={\cal K}_{k,j_1,\dots,j_k}(G_{j_1,j_1},\dots,G_{j_k,j_k})$
depending on the diagonal elements $G_{1,1},\dots,G_{d,d}$
of a matrix valued spectral measure $(G_{j,j'})$, 
$1\le j,j'\le d$, and a sequence of length $k$ of 
integers $(j_1,\dots,j_k)$ such that $1\le j_s\le d$ 
for all $1\le s\le k$. (In paper~\cite{4} I worked with 
Wiener--It\^o integrals of order~$n$, while here I work 
with Wiener--It\^o integrals of order~$k$. Hence I use 
here a slightly different notation.) I shall refer to 
this Hilbert space in the formulation of our result. 
This Hilbert space appears in this result, because it 
contains those functions $f(x_1,\dots,x_k)$ for 
which we defined the $k$-fold Wiener--It\^o integral 
$\int f(x_1,\dots,x_k)Z_{G,{j_1}}(\,dx_1)\dots Z_{G,{j_k}}(\,dx_k)$
with respect to a vector valued random spectral measure
$(Z_{G,1},\dots,Z_{G,d})$ corresponding to the matrix 
valued spectral measure $(G_{j,j'})$, $1\le j,j'\le d$.

\medskip
We take for all $N=1,2,\dots$ 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 a parameter~$A_N$ such that $A_N\to\infty$ as 
$N\to\infty$. We also take some functions 
$$
h^N_{j_1,\dots,j_k}(x_1,\dots,x_k)\in {\cal K}_{k,j_1,\dots,j_k}
={\cal K}_{k,j_1,\dots,j_k}(G^{(N)}_{j_1,j_1},\dots,G^{(N)}_{j_k,j_k})
$$
on the torus $[-A_N\pi,A_N\pi)^\nu$ for all sequences
$(j_1,\dots,j_k)$ such that $1\le j_s\le d$, 
$1\le s\le k$, and $N=1,2,\dots$. Besides, we fix for all 
$N=1,2,\dots$ a vector valued random spectral measure 
$(Z_{G^{(N)},1},\dots,Z_{G^{(N)},d})$ on the torus 
$[-A_N\pi,A_N\pi)^\nu$ corresponding to the matrix valued 
spectral measure $(G^{(N)}_{j,j'})$, $1\le j,j'\le d$, and 
we define with the help of these quantities the  
sums of $k$-fold Wiener--It\^o integrals
\begin{equation}
S_N=\sum_{\substack{(j_1,\dots,j_k)\\ 
1\le j_s\le d, \textrm{ for all }1\le s\le k}}
\int h^N_{j_1,\dots,j_k}(x_1,\dots,x_k)Z_{G^{(N)},{j_1}}(\,dx_1)
\dots Z_{G^{(N)},{j_k}}(\,dx_k),    \label{2.6}
\end{equation}
$N=1,2,\dots$. We want to find some good conditions 
under which these random variables $S_N$ converge
in distribution to a random variable $S_0$ expressed
in the form of a sum of multiple Wiener--It\^o integrals.

The following result supplies such conditions.

\medskip\noindent
{\bf Proposition 2A.} {\it Let us consider the sums of
$k$-fold Wiener--It\^o integrals $S_N$ defined in 
formula~(\ref{2.6}) with the help of certain vector 
valued random spectral measures 
$(Z_{G^{(N)},1},\dots,Z_{G^{(N)},d})$ corresponding to some 
non-atomic matrix valued spectral measures 
$(G^{(N)}_{j,j'})$, $1\le j,j'\le d$ defined on 
tori $[-A_N,A_N)^\nu$ such that $A_N\to\infty$ as 
$N\to\infty$ and functions 
$$
h^N_{j_1,\dots,j_k}(x_1,\dots,x_k)\in 
{\cal K}_{k,j_1,\dots,j_k}(G^{(N)}_{j_1,j_1},\dots,G^{(N)}_{j_k,j_k}).
$$
Let the coordinates $G^{(N)}_{j,j'}$, $1\le j,j'\le d$, of the 
matrix valued spectral measures $(G^{(N)}_{j,j'})$, $1\le j,j'\le d$, 
converge vaguely to the coordinates $G^{(0)}_{j,j'}$ of a non-atomic 
matrix valued spectral measure $(G^{(0)}_{j,j'})$, $1\le j,j'\le d$, on 
${\mathbb R}^\nu$ for all $1\le j,j'\le d$ as $N\to\infty$, and let 
$(Z_{G^{(0)},1},\dots,Z_{G^{(0)},d})$ be a vector valued random 
spectral measure on ${\mathbb R}^\nu$ corresponding to the matrix
valued spectral measure $(G^{(0)}_{j,j'})$, $1\le j,j'\le d$. 
Let us also have some functions $h^0_{j_1,\dots,j_k}$
for all $1\le j_s\le d$, $1\le s\le k$, such that 
these functions and matrix valued spectral measures 
satisfy the following conditions.

\medskip
\begin{description}
\item[\rm{(a)}] The functions $h^0_{j_1,\dots,j_k}(x_1,\dots,x_k)$ 
are continuous on ${\mathbb R}^{k\nu}$ for all $1\le j_s\le d$, 
$1\le s\le k$,  and for all $T>0$ and indices 
$1\le j_s\le d$, $1\le s\le k$, the functions 
$h^N_{j_1,\dots,j_k}(x_1,\dots,x_k)$ converge uniformly to 
the function $h^0_{j_1,\dots,j_k}(x_1,\dots,x_k)$ on the 
cube $[-T,T]^{k\nu}$ as $N\to\infty$.
\item[\rm{(b)}] For all $\varepsilon>0$ there is some 
$T_0=T_0(\varepsilon)>0$ such that 
\begin{equation}
\int_{{\mathbb R}^{k\nu}\setminus [-T,T]^{k\nu}} |h^N_{j_1,\dots,j_k}(x_1,\dots,x_k)|^2
G^{(N)}_{j_1,j_1}(\,dx_1)\dots G^{(N)}_{j_k,j_k}(dx_k)<\varepsilon^2 
\label{2.7} 
\end{equation}
for all $1\le j_s\le d$, $1\le s\le k$, and $N=1,2\dots$ if $T>T_0$.
\end{description}

\medskip
Then inequality~(\ref{2.7}) holds also for $N=0$, 
$$
h^0_{j_1,\dots,j_k}(x_1,\dots,x_k)\in{\cal K}_{k,j_1,\dots,j_k}
={\cal K}_{k,j_1,\dots,j_k}(G^{(0)}_{j_1,j_1},\dots G^{(0)}_{j_k,j_k}),
$$
the sum of random integrals 
\begin{equation}
S_0=\sum_{\substack{(j_1,\dots,j_k)\\ 
1\le j_s\le d, \textrm{ for all }1\le s\le k}}
\int h^0_{j_1,\dots,j_k}(x_1,\dots,x_k)Z_{G^{(0)},{j_1}}(\,dx_1)
\dots Z_{G^{(0)},{j_k}}(\,dx_k). \label{2.8}
\end{equation}
exists, and the random variables $S_N$ defined in~(\ref{2.6}) 
converge to $S_0$ in distribution as $N\to\infty$.}

\medskip
I shall prove Theorem~1.2A with the help of Proposition~2A
and the representation of the random variables $S_N$, 
$N=1,2,\dots$, in formulas~(\ref{2.1}) and~(\ref{2.2}). I shall 
apply Proposition~2A with an appropriate choice of the random 
variables $S_N$, $N=1,2,\dots$, in formula (\ref{2.6}) and of the 
random variable $S_0$ in~(\ref{2.8}). In this application I define 
the random variables $S_N$ in (\ref{2.6}) and the random variable
$S_0$ in~(\ref{2.8}) by rewriting first the random variables 
defined in (\ref{2.1}) and (\ref{2.2}) and then the random 
variable defined in (\ref{2.4}) and (\ref{2.5}) in an appropriate 
way. I shall rewrite them in such a special form of 
formulas~(\ref{2.6}) and~(\ref{2.8}) where the summation is going 
only for such sequences $(j_1,\dots,j_k)$ whose elements 
$j_s$, $1\le s\le k$, go up in increasing order from $1$ to~$d$, 
i.e. $1\le j_1\le j_2\le\cdots\le j_k\le d$. 
Given a sequence $(j_1,\dots,j_k)$ with this property 
I define with its help the numbers
\begin{eqnarray}
k_s(j_1,\dots j_k)&=&\textrm{the number of such elements } j_p 
\nonumber \\ 
&&\qquad
\textrm{in the sequence }j_1,\dots,j_k \textrm{ for which }
j_p=s \label{2.9}
\end{eqnarray}
for all indices $1\le s\le d$. 

I shall rewrite the sums in (\ref{2.1}) and (\ref{2.5}) with 
which I shall work by replacing the indices of summation 
$k_1,\dots,k_d$ in them by appropriately chosen indices 
$1\le j_1\le\dots\le j_k\le d$. I choose these  
indices $j_1,\dots,j_k$ in such a way that the 
quantities $k_s=k_s(j_1,\dots,j_k)$, $1\le s\le d$,
defined in~(\ref{2.9}) agree with the original sequence
$(k_1,\dots,k_d)$ in~(\ref{2.1}) or~(\ref{2.5}).
I exploit that there is a natural one to one 
correspondence between the sequences $j_1,\dots,j_k$ 
such that $1\le j_1\le\dots\le j_k\le d$ and the
sequences $k_1,\dots,k_d$ of integers with the property
$k_s\ge0$ for all $1\le s\le d$ and $k_1+\cdots+k_d=k$.
With the help of this correspondence the 
random sums $S_N$ in~(\ref{2.1}) can be rewritten in the form
\begin{eqnarray}
S_N&=& \sum_{\substack{(j_1,\dots,j_k),\\ 
1\le j_1\le \cdots\le j_k\le d}}
 \int f^N_{k_1(j_1,\dots,j_k),\dots,k_d(j_1,\dots,j_k)}(x_1,\dots,x_k) 
\nonumber \\
&&\qquad\qquad\qquad\qquad Z_{G^{(N)},{j_1}}(\,dx_1)\dots 
Z_{G^{(N)},{j_k}}(\,dx_k) \label{2.10}
\end{eqnarray}
for all $N=1,2,\dots$ with the functions
$f^N_{k_1,\dots,k_d}(x_1,\dots,x_k)$ defined in~(\ref{2.2}) and
the functions $k_s(j_1,\dots,j_k)$, $1\le s\le d$ defined
in~(\ref{2.9}). This means that relation~(\ref{2.10}) can
be rewritten in the form 
\begin{equation}
S_N=\sum_{\substack{(j_1,\dots,j_k),\\ 1\le j_1\le \cdots\le j_k\le d}}
 \int h^{N}_{j_1,\dots,j_k}(x_1,\dots,x_k) 
Z_{G^{(N)},{j_1}}(\,dx_1)\dots 
Z_{G^{(N)},{j_k}}(\,dx_k) \label{2.11}
\end{equation}
with
\begin{equation}
 h^{N}_{j_1,\dots,j_k}(x_1,\dots,x_k)= 
f^N_{k_1(j_1,\dots,j_k),\dots,k_d(j_1,\dots,j_k)}(x_1,\dots,x_k), \label{2.12} 
\end{equation}
where the functions $k_s(j_1,\dots,j_k)$, $1\le s\le d$, are
defined in (\ref{2.9}).
Similarly, the random sum $S_0$ in~(\ref{2.5}) can 
be rewritten in the form
\begin{eqnarray*}
S_0&=&\sum_{\substack{(j_1,\dots,j_k)\\ 
1\le j_1\le \cdots\le j_k\le d}}
\int f^0_{k_1(j_1,\dots,j_k),\dots,k_d(j_1,\dots,j_k)}(x_1,\dots,x_k)  \\
&&\qquad\qquad\qquad\qquad Z_{G^{(0)},{j_1}}(\,dx_1)\dots 
Z_{G^{(0)},{j_k}}(\,dx_k) 
\end{eqnarray*}
with the function $f^0_{k_1,\dots,k_d}(x_1,\dots,x_k)$ defined
in~(\ref{2.4}) or in the following equivalent form
\begin{equation}
S_0=\sum_{\substack{(j_1,\dots,j_k),\\ 1\le j_1\le \cdots\le j_k\le d}}
 \int h^{0}_{j_1,\dots,j_k}(x_1,\dots,x_k) 
Z_{G^{(0)},{j_1}}(\,dx_1)\dots 
Z_{G^{(0)},{j_k}}(\,dx_k) \label{2.13}
\end{equation}
with
\begin{equation}
 h^{0}_{j_1,\dots,j_k}(x_1,\dots,x_k)= 
f^0_{k_1(j_1,\dots,j_k),\dots,k_d(j_1,\dots,j_k)}(x_1,\dots,x_k). \label{2.14} 
\end{equation}

\medskip
I shall apply Proposition~2A for the sequences 
$S_N$, defined in~(\ref{2.11}), (\ref{2.2}) and~(\ref{2.12}) 
for $N=1,2,\dots$, and in~(\ref{2.13}), (\ref{2.4}) 
and~(\ref{2.14}) for $N=0$. I shall integrate with 
respect to the coordinates of vector valued 
spectral measures $(Z_{G^{(N)},1},\dots,Z_{G^{(N)},d})$, 
$N=0,1,2,\dots$, corresponding to the matrix valued 
spectral measures $(G^{(N)}_{j.j'})$, $1\le j,j'\le d$, 
$N=0,1,2,\dots$, defined in formula~(\ref{1.5}) for $N\ge1$ 
and in Proposition~1.1 for $N=0$. In the proof of 
Theorem~1.2A I have to show that the conditions of 
Proposition~2A are satisfied with such a choice.

It follows from Proposition~1.1 that the (non-atomic)
complex measures $G^{(N)}_{j,j'}$ with finite
total variation vaguely converge to the (non-atomic) 
complex measure $G^{(0)}_{j,j'}$ with 
locally finite total variation as $N\to\infty$ for 
all $1\le j,j'\le d$. It is also clear that
$h^N_{j_1,\dots,j_k}(x_1,\dots,x_k)\in 
{\cal K}_{k,j_1,\dots,j_k}(G^{(N)}_{j_1,j_1},\dots,G^{(N)}_{j_k,j_k})$
for all $1\le j_1\le\cdots \le j_k\le d$ and $N=1,2,\dots$ 
with $h^N_{j_1,\dots,j_k}(x_1,\dots,x_k)$ defined in~(\ref{2.12}). 

It follows from (\ref{2.3}), (\ref{2.12}) and~(\ref{2.14}) 
that condition~(a) of Proposition~2A holds with the choice 
of the functions and measures we chose in the proof of 
Proposition~1.2.  We still have to prove relation~(\ref{2.7}) 
in condition~(b). This will be done with the help of the 
following Proposition~2.1. (Actually we shall prove in 
Proposition~2.1 a slightly stronger result than we need.)

\medskip\noindent
{\bf Proposition 2.1.} {\it Let us fix some integer $k\ge1$,
and let $(G_{j,j'})$, $1\le j,j'\le d$, be the matrix valued 
spectral measure of a vector valued stationary field 
$X(p)=(X_1(p),\dots,X_p(d))$, $p\in{\mathbb Z}^\nu$, 
defined on the torus $[-\pi,\pi)^\nu$, whose correlation 
function $r_{j,j'}(p)=EX_j(0)X_{j'}(p)$, $1\le j,j'\le d$, 
$p\in{\mathbb Z}^\nu$, satisfies relation~(\ref{1.2}) with some 
$0<\alpha<\frac1k$. For all $N=1,2,\dots$, let us consider 
the measures $G^{(N)}_{j,j}$, $1\le j\le d$, defined in 
formula~(\ref{1.5}) together with the measures 
$\mu^{(N)}_{j_1,\dots,j_k}$ defined for all sets of indices
$j_1,\dots,j_k$, $1\le j_s\le d$, $1\le s\le k$, on 
${\mathbb R}^{k\nu}$ by the formula
\begin{equation}
\mu^{(N)}_{j_1,\dots,j_k}(A)=\int_A|h_N(x_1,\dots,x_k)|^2 
G^{(N)}_{j_1,j_1}(\,dx_1)\dots G^{(N)}_{j_k,j_k}(\,dx_k), 
\quad A\in {\cal B}^{k\nu}, \label{2.15}
\end{equation}
with
\begin{equation}
h_N(x_1,\dots,x_k)=\prod_{l=1}^\nu
\frac{e^{i((x_1^{(l)}+\cdots+x_k^{(l)})}-1} 
{N(e^{i((x_1^{(l)}+\cdots+x_k^{(l)})/N}-1)}, \label{2.16} 
\end{equation}
where we use the notation $x=(x^{(1)},\dots,x^{(\nu)})$ 
for a vector $x\in {\mathbb R}^\nu$. The measures 
$\mu^{(N)}_{j_1,\dots,j_k}$ converge weakly to a finite 
measure $\mu^{(0)}_{j_1,\dots,jk}$ on ${\mathbb R}^{k\nu}$.} 

\medskip\noindent
{\it Proof of Theorem 1.2A with the help of Proposition 2.1.}
It is enough to show that the complex measures  $G^{(N)}_{j,j'}$,
$1\le j,j'\le d$, $N=0,1,2,\dots$, defined in~(\ref{1.5}) and
in the result of Proposition~1.1 together with the functions
$h^N_{j_1,\dots,j_k}$, $1\le j_s,\le d$, for all $1\le s\le k$ and
$N=0,1,2,\dots$ defined in (\ref{2.2}), (\ref{2.11})
and~(\ref{2.12}) for $N=1,2,\dots$ and in~(\ref{2.4}), (\ref{2.13})
and~(\ref{2.14}) for $N=0$ satisfy the conditions of Proposition~2A.
We have proved the validity of all these conditions expect
formula~(\ref{2.7}) in condition~(b). But even this condition
is simply follows from Proposition~2.1 which implies that the
measures $\mu^{(N)}_{j_1,\dots,j_k}$, $N=1,2,\dots$, are tight for all
indices $N=1,2,\dots$. This fact together with the definition
of the measures $\mu^{(N)}_{j_1,\dots,j_k}$ and the identity
$h^N_{j_1,\dots,j_k}(x_1,\dots,x_k)=
c_{k_1(j_1,\dots,j_k),\dots,k_d(j_1,\dots,j_k)} h_N(x_1,\dots,x_k)$
imply that relation~(\ref{2.7}) holds. Theorem~1.2A is proved. 

\medskip
It remained to prove Proposition~2.1.

\medskip\noindent
{\it Proof of Proposition 2.1.} Most calculations needed
in the proof of Proposition 2.1 were actually carried out
in the proof of Theorem~8.2 of~\cite{3} with some slight 
modifications. I shall omit the details of these 
calculations. 

I compute for all $N=1,2,\dots$ the Fourier transform
$$
\varphi^{(N)}_{j_1,\dots,j_d}(t_1,\dots,t_k)=\int e^{i((t_1,x_1)+\cdots+(t_k,x_k))}
\mu^{(N)}_{j_1,\dots,j_k}(\,dx_1,\dots,dx_k)
$$
of the measures $\mu^{(N)}_{j_1,\dots,j_k}$ defined in~(\ref{2.15}), 
and give a good asymptotic formula for them. More precisely, 
I do this only for such coordinates $(t_1,\dots,t_k)$ of 
the function $\varphi^{(N)}_{j_1,\dots,j_d}(t_1,\dots,t_k)$ 
which have the form $t_l=\frac {p_l}N$ with some 
$p_l\in {\mathbb Z}^\nu$, $l=1,\dots,k$. But even such a 
result will be sufficient for us. In the calculation of the 
formula expressing 
$\varphi^{(N)}_{j_1,\dots,j_d}(t_1,\dots,t_k)$ I exploit that
the function $h_N(x_1,\dots,x_k)$ defined in~(\ref{2.16}) can be 
written in the form
$$
h_N(x_1,\dots,x_k)=\frac1{N^\nu}\sum_{u\in B_N}
\exp\left\{i\frac1N(u,x_1+\cdots+x_k)\right\}.
$$
Hence 
\begin{eqnarray*}
&&\!\!\!\!\!\!
\varphi^{(N)}_{j_1,\dots,j_d}(t_1,\dots,t_k)
=\frac1{N^{2\nu}}\int\exp\left\{
i\frac1N((p_1,x_1)+\cdots+(p_k,x_k))\right\}    \\
&&\qquad\qquad\sum_{u\in B_N}\sum_{v\in B_N}
\exp\left\{i\left(\frac{u-v}N,x_1+\cdots+x_k\right)\right\} \\
&& \qquad\qquad\qquad\qquad
G^{(N)}_{j_1,j_k}(\,dx_1)\dots G^{(N)}_{j_k,j_k}(\,dx_k)   \\
&&\qquad=\frac1{N^{2\nu-k\alpha}L(N)^k}\sum_{u\in B_N}\sum_{v\in B_N}
r_{j_1,j_1}(u-v+p_1)\cdots r_{j_k,j_k}(u-v+p_k)
\end{eqnarray*}
if $t_l=\frac {p_l}N$ with some $p_l\in{\mathbb Z}^\nu$, 
$1\le l\le k$. This identity can be rewritten by taking 
the summation at the right-hand side of the last formula 
first for such pairs of $(u,v)$ for which $u-v=y$ with 
some fixed value $y\in{\mathbb Z}^\nu$ and then for 
the lattice points $y\in{\mathbb Z}^\nu$. By working
with $x=\frac yN$ instead of $y$ we get that  
$$
\varphi^{(N)}_{j_1,\dots,j_d}(t_1,\dots,t_k)
=\int_{[-1,1]^\nu} f^{(N)}_{j_1,\dots,j_k}(t_1,\dots,t_k,x)\lambda_N(\,dx)
$$
with
\begin{eqnarray*}
&& \!\!\!\!\!\!\!\!\!\!\!\!\!
f^{(N)}_{j_1,\dots,j_k}(t_1,\dots,t_k,x) \\
&&\!\!=\left(1-\frac{|x^{(1)}N|}N\right)\cdots
\left(1-\frac{|x^{(\nu)}N|}N\right)
\frac{r_{j_1,j_1}(N(x+t_1))}{N^{-\alpha}L(N)}\cdots
\frac{r_{j_k,j_k}(N(x+t_k))}{N^{-\alpha}L(N)},
\end{eqnarray*}
where $\lambda_N$ is the measure concentrated in the points of 
the form $x=\frac pN$ with such points
$p=(p_1,\dots,p^\nu)\in {\mathbb Z}^\nu$ for which $-N<p_l<N$
for all $1\le l\le\nu$, and $\lambda_N(x)=N^{-\nu}$ for such 
points~$x$.

Let us extend the definition of
$\varphi^{(N)}_{j_1,\dots,j_d}(t_1,\dots,t_k)$ to 
$(t_1,\dots,t_k)\in{\mathbb R}^{k\nu}$ by defining it as
$$
\varphi^{(N)}_{j_1,\dots,j_d}(t_1,\dots,t_k)
=\varphi^{(N)}_{j_1,\dots,j_d}\left(\frac{p_1}N,\dots,\frac{p_k}N\right),
\quad t_l\in{\mathbb R}^\nu \textrm{ for all } 1\le l\le k,
$$ 
if $\frac{p_l^{(s)}}N-\frac1{2N}\le t_l^{(s)}<\frac{p_l^{(s)}}N+\frac1{2N}$
with $p_l=(p^{(1)}_l,\dots,p^{(\nu)}_l)\in{\mathbb Z}^\nu$, $1\le l\le k$. 
Let us extend similarly the definition of the function
$f^{(N)}_{j_1,\dots,j_k}(t_1,\dots,t_k,x)$ to 
$(t_1,\dots,t_k,x)\in{\mathbb R}^{k\nu}\times[-1,1]^\nu$ by means 
of the formula
\begin{eqnarray*}
&& \!\!\!\!\!\!\!\!\!\!\!\!\!
f^{(N)}_{j_1,\dots,j_k}(t_1,\dots,t_k,x) \\ 
&&\!\!=\left(1-\frac{|q^{(1)}N|}N\right)\cdots
\left(1-\frac{|q^{(\nu)}N|}N\right)
\frac{r_{j_1,j_1}(N(q+p_1))}{N^{-\alpha}L(N)}\cdots
\frac{r_{j_k,j_k}(N(q+p_k))}{N^{-\alpha}L(N)},
\end{eqnarray*}
for $t_l\in{\mathbb R}^\nu$, $1\le l\le k$, and $x\in[-1,1]^\nu$,
where
$\frac{p_l^{(s)}}N-\frac1{2N}\le t_l^{(s)}<\frac{p_l^{(s)}}N+\frac1{2N}$,
$1\le s\le\nu$, $1\le l\le k$, 
with $p_l=(p^{(1)}_l,\dots,p^{(\nu)}_l)\in{\mathbb Z}^\nu$, 
$1\le l\le k$, and 
$\frac{q^{(s)}}N-\frac1{2N}\le x^{(s)}<\frac{q^{(s)}}N+\frac1{2N}$,
$1\le s\le\nu$ with $q=(q^{(1)},\dots,q^{(\nu)})\in{\mathbb Z}^\nu$.

It follows from relation~(\ref{1.2}) that for all parameters
$t_1,\dots,t_k$ and $\varepsilon>0$
$$
f^{(N)}_{j_1,\dots,j_k}(t_1,\dots,t_k,x)
\to f^{(0)}_{j_1,\dots,j_k}(t_1,\dots,t_k,x) 
$$
holds uniformly with the limit function
\begin{eqnarray*}
&&f^{(0)}_{j_1,\dots,j_k}(t_1,\dots,t_k,x) \\
&&\qquad =(1-|x^{(1)}|)\dots(1-|x^{(\nu)}|)
\frac{a_{j_1,j_1}\left(\frac{x+t_1}{|x+t_1|}\right)}{|x+t_1|^\alpha}\dots
\frac{a_{j_k,j_k}\left(\frac{x+t_k}{|x+t_k|}\right)}{|x+t_k|^\alpha}
\end{eqnarray*}
on the set $x\in[-1,1]^\nu \setminus
\bigcup\limits_{l=1}^k\{x\colon\;|x+t_l|>\varepsilon\}$. 

Some additional calculation shows that
\begin{equation}
\varphi^{(N)}_{j_1,\dots,j_k}(t_1,\dots,t_k)\to
\varphi^{(0)}_{j_1,\dots,j_k}(t_1,\dots,t_k)
=\int_{[-1,1]^\nu}f^{(0)}_{j_1,\dots,j_k}(t_1,\dots,t_k,x)\,dx
\label{2.17}
\end{equation}
as $N\to\infty$ for all $(t_1,\dots,t_k)\in {\mathbb R}^{k\nu}$, 
and $\varphi^{(0)}_{j_1,\dots,j_k}(t_1,\dots,t_k)$ is a 
continuous function. This calculation was carried out 
in the proof of Theorem~8.2 in~\cite{3} (in the calculations
made after the end of the proof of Lemma~8.4), hence I omit it.

By a classical result of probability theory if the Fourier 
transforms of a sequence of finite measures on ${\mathbb R}^{k\nu}$
converge to a function continuous at the origin, then the
limit function is also the Fourier transform of a finite
measure on ${\mathbb R}^{k\nu}$, and the sequence of probability 
measures whose Fourier transforms were taken converge to 
this measure. In the proof of Theorem~2.1 we cannot apply 
this result, because we have a control on the Fourier 
transform of $\mu^{(N)}_{j_1,\dots,j_k}$ only in points of 
the form $(t_1,\dots,t_k)$ with $t_l=\frac{p_l}N$ with 
$p_l\in{\mathbb Z}^\nu$, $1\le l\le k$. But the measures
$\mu^{(N)}_{j_1,\dots,j_k}$ have the additional property that
they are concentrated in the cube $[-N\pi,N\pi)^{k\nu}$.
Lemma~8.4 of~\cite{3} contains such a result that enables us
to deduce from relation~(\ref{2.17}) (together with the 
continuity of the limit function 
$\varphi^{(0)}_{j_1,\dots,j_k}(t_1,\dots,t_k)$) and the above 
mentioned property of the measures $\mu^{(N)}_{j_1,\dots,j_k}$ 
the convergence of the measures~$\mu^{(N)}_{j_1,\dots,j_k}$ to 
a finite measure $\mu^{(0)}_{j_1,\dots,j_k}$. We have also
proved that this finite measure $\mu^{(0)}_{j_1,\dots,j_k}$,
has the Fourier transform $\varphi^{(0)}(t_1,\dots,t_k)$. 
Proposition~2.1 is proved.

\medskip
I prove Theorem~1.2B by showing that if a function 
$H^{(1)}(\cdot)$ satisfies (\ref{1.8}) and (\ref{1.9}), and the
Gaussian stationary process $Z(p)=(X_1(p),\dots,X_d(p))$ 
satisfies~(\ref{1.2}) and~(\ref{1.3}), then 
\begin{equation}
\frac1{N^{\nu-k\alpha/2}L(N)^{k/2}}\sum_{p\in B_N}H^{(1)}(X_1(p),\dots,X_d(p))
\Rightarrow0 \quad \textrm{as }N\to\infty, \label{2.18}
\end{equation}
where $\Rightarrow$ denotes convergence in probability. I 
shall prove that even the second moments of the normalized 
sums in(\ref{2.18}) tend to zero. The following Lemma~2B 
which agrees with Lemma~1 of~\cite{1} (only with a slightly 
different notation) helps in the proof of this statement.

\medskip\noindent
{\bf Lemma 2B.} {\it Let $X_1,\dots,X_d$ and $Y_1,\dots Y_d$
be jointly Gaussian random variables with expectation zero 
such that $EX_jX_{j'}=EY_jY_{j'}=\delta_{j,j'}$, $1\le j,j'\le d$, 
and let $r_{j,j'}=EX_jY_{j'}$, $1\le j,j'\le d$. Take a number 
$k\ge1$ and a function $H^{(1)}(x_1,\dots,x_d)$ satisfying 
relations~(\ref{1.8}) and~(\ref{1.9}). Assume that
$$
\psi=\max\left( \left(\sup_{1\le j\le d}\sum_{j'=1}^d |r_{j,j'}|\right),
\left(\sup_{1\le j'\le d}\sum_{j=1}^d |r_{j,j'}|\right)\right)\le 1.
$$
Then
$$
|EH^{(1)}(X_1,\dots,X_d)H^{(1)}(Y_1,\dots,Y_d)|\le \psi^{k+1}
E{H^{(1)}}^2(X_1,\dots,X_d).
$$
}

\medskip\noindent
{\it Proof of Theorem 1.2.} It follows from relation (\ref{1.2}) and
Lemma~2B with the choice $X_j=X_j(p)$, $Y_j=X_j(q)$, $1\le j\le d$, 
that there exists some threshold index $n_0\ge1$ and constant 
$0<C<\infty$ such that 
$$
|EH^{(1)}(X_1(p),\dots X_d(p))H^{(1)}(X_1(q),\dots,X_d(q))|
\le C|p-q|^{-(k+1)\alpha}L(|p-q|)^{k+1}
$$ 
if $|p-q|\ge n_0$. On the other hand,
\begin{eqnarray*}
&&|EH^{(1)}(X_1(p),\dots X_d(p))H^{(1)}(X_1(q),\dots,X_d(q))|\\
&&\qquad \qquad \le E{H^{(1)}}^2(X_1(0),\dots X_d(0))\le C_1
\end{eqnarray*}
for all $p,q\in{\mathbb Z}^\nu$ with some $C_1<\infty$ by the 
Schwarz inequality and relation~(\ref{1.9}). Hence
\begin{eqnarray*}
&&\left|EH^{(1)}(X_1(p),\dots,X_d(p))
\left(\sum_{q\in B_N}H^{(1)}(X_1(q),\dots,X_d(q)\right)\right| \\
&&\qquad\qquad\qquad \le C_2(1+N^{\nu+\varepsilon-(k+1)\alpha})
\end{eqnarray*}
for all $p\in B_N$ and $\varepsilon>0$ with an appropriate 
$C_2=C_2(\varepsilon)>0$, and since 
$\nu-k\alpha>0$
$$
\frac1{N^{2\nu-k\alpha}L(N)^k}
E\left[\sum_{p\in B_N}H^{(1)}(X_1(p),\dots,X_d(p))\right]^2\to0 
\textrm\quad \textrm{as }N\to\infty.
$$
The last relation implies formula~(\ref{2.18}). 
Formula~(\ref{2.18}) together Theorem~1.2A yield 
Theorem~1.2B. Theorem~1.2B is proved.

\medskip
The proof of Theorem 1.3 is very similar to that of Theorems~1.2A
and~1.2B, hence I shall explain it briefly.

It is enough to prove that for any positive integer $K$, 
parameters $t_1,\dots,t_K$, $t_p\in[0,1]^\nu$, 
$1\le p\le K$ and real constants $C_1,\dots,C_K$ the linear 
combinations $\sum_{p=1}^K C_p S_N(t_p)$ converge to
$\sum_{p=1}^K C_p S_0(t_p)$ in distribution as $N\to\infty$,
since this implies that the random vectors 
$(S_N(t_1),\dots,S_N(t_K))$ converge in distribution to 
the random vector $(S_0(t_1),\dots,S_0(t_K))$ as 
$N\to\infty$. Moreover, similarly to the proof of 
Theorem~1.2B we can reduce the proof of the result to 
the case when $H(x_1,\dots,x_d)=H^{(0)}(x_1,\dots,x_d)$ 
with a function $H^{(0)}(x_1,\dots,x_d)$ of the form 
given in~(\ref{1.4}).

In the first step of the proof I write the linear 
combinations $\sum_{p=1}^K C_p S_N(t_p)$, $N=0,1,2,\dots$, 
in the form of a sum of $k$-fold Wiener-It\^o integrals 
with respect to the coordinates of an appropriate vector 
valued random spectral measure. 
We can do this if we can write down the random variables 
$S_N(t)$ in the desired form for all $t\in[0,1]^\nu$. The 
random variables $S_0(t)$ are given in this form in~(\ref{1.12}). 
In the case $N=1,2,\dots$ we can find the right 
formulation of $S_N(t)$ similarly to the method applied 
in the proof of Theorem~1.2A. We get similarly to the
proof of formulas~(\ref{2.1}) and~(\ref{2.2}) that 
\begin{eqnarray*}
S_N(t)&=& 
\sum_{\substack{(k_1,\dots,k_d),\; k_j\ge0,\;1\le j\le d,\\ 
k_1+\cdots +k_d=k}}
 \int f^N_{k_1,\dots,k_d}(t,x_1,\dots,x_k) \\
&&\qquad\qquad\qquad \prod_{j=1}^d 
\left(\prod_{t=k_1+\cdots+k_{j-1}+1}^{k_1+\cdots+k_j} 
Z_{G^{(N)},j}(\,dx_t)\right)
\end{eqnarray*}
with
$$
f^N_{k_1,\dots,k_d}(t,x_1,\dots,x_k)=c_{k_1,\dots,k_d}\prod_{l=1}^\nu 
\frac{\exp\left\{i\frac{]t^{(l)}N[}N(x_1^{(l)}+\cdots+x_k^{(l)})\right\}-1} 
{N\left(\exp\left\{i\frac1N(x_1^{(l)}+\cdots+x_k^{(l)})\right\}-1\right)}, 
$$
where $t=(t^{(1)},\dots,t^{\nu})$, the number $]t^{(l)}N[$ (in the
definition of the function \hfill\break 
$f_{k_1,\dots,k_d}^N(t,x_1,\dots,x_k)$) 
is the smallest integer which is larger than $t^{(l)}N$, and 
$Z_{G^{(N)},j}$ agrees with the spectral measure that appeared 
in~(\ref{2.1}) under the same notation.

It is not difficult to see that similarly to relations 
(\ref{2.3}) and (\ref{2.4})
$$
\lim_{N\to\infty} f^N_{k_1,\dots,k_d}(t,x_1,\dots,x_k)
=f^0_{k_1,\dots,k_d}(t,x_1,\dots,x_k) 
$$
for all indices $k_1,\dots, k_d$ such that $k_j\ge0$, $1\le j\le d$,
and $k_1+\cdots+k_d=k$ with the function
$$
f^0_{k_1,\dots,k_d}(t,x_1,\dots,x_k)=c_{k_1,\dots,k_d}\prod_{l=1}^\nu
\frac{e^{it^{(l)}(x_1^{(l)}+\cdots+x_k^{(l)})}-1} 
{i(x_1^{(l)}+\cdots+x_k^{(l)})}, 
$$
defined for $(x_1.\dots,x_k)\in{\mathbb R}^{k\nu}$, and this 
convergence is uniform in all bounded subsets of 
${\mathbb R}^{k\nu}$.

With the help of the above considerations the proof of 
Theorem~1.3 can be reduced, similarly to the proof of 
Theorem~1.2A, to the following statement. Fix some 
number~$K$, real constants $C_1,\dots,C_K$ and points 
$t_1,\dots t_K$ with $t_p\in [0,1]^\nu$, $1\le p\le K$ 
together with some constants $c_{k_1,\dots,k_d}$ with parameters 
$k_j\ge0$, $1\le j\le d$, and $k_1+\cdots+k_d=k$ as the 
coefficients in the sum~(\ref{1.4}). Let us define with 
their help the random sums
\begin{eqnarray}
S_N&=& 
\sum_{\substack{(k_1,\dots,k_d),\; k_j\ge0,\;1\le j\le d,\\ 
k_1+\cdots +k_d=k}} \int \left(\sum_{p=1}^K
C_pc_{k_1,\dots,k_d} f^N_{k_1,\dots,k_d}(t_p,x_1,\dots,x_k)\right) \nonumber \\ 
&&\qquad\qquad\qquad \prod_{j=1}^d 
\left(\prod_{t=k_1+\cdots+k_{j-1}+1}^{k_1+\cdots+k_j} 
Z_{G^{(N)},j}(\,dx_t)\right) \label{2.19} 
\end{eqnarray}
with the above defined functions $g^{(N)}_{k_1,\dots,k_d}(t,x_1,\dots,x_k)$
for all $N=1,2,\dots$, and
\begin{eqnarray}
S_0&=& 
\sum_{\substack{(k_1,\dots,k_d),\; k_j\ge0,\;1\le j\le d,\\ 
k_1+\cdots +k_d=k}} 
\int \left(\sum_{p=1}^K
C_pc_{k_1,\dots,k_d} f^0_{k_1,\dots,k_d}(t_p,x_1,\dots,x_k)\right) \nonumber \\ 
&&\qquad\qquad\qquad \prod_{j=1}^d 
\left(\prod_{t=k_1+\cdots+k_{j-1}+1}^{k_1+\cdots+k_j} 
Z_{G^{(0)},j}(\,dx_t)\right) \label{2.20} 
\end{eqnarray}
with the previously defined function 
$g^0_{k_1,\dots,k_d}(t,x_1,\dots,x_k)$. I claim that the sequence of 
random variables $S_N$ defined in~(\ref{2.19}) converge in 
distribution to $S_0$ defined in~(\ref{2.20}) as $N\to\infty$.

\medskip
This statement can be proved similarly to Theorem~1.2A 
with the help of Proposition~2A. In this proof we rewrite 
the random variables $S_N$, $N=1,2,\dots$, and $S_0$ in 
a form in which we can apply Proposition~2A. We rewrite 
them as a sum of multiple Wiener--It\^o integrals indexed 
by sequences of integers $j_1,\dots,j_k$ such that
$1\le j_1\le\cdots\le j_k\le d$. This can be done 
similarly to the reformulation of formulas~(\ref{2.1}) 
and~(\ref{2.5}) in formulas~(\ref{2.11}), (\ref{2.12}) 
and~(\ref{2.13}). (\ref{2.14}) with the help
of the expressions $k_s(j_1\dots,j_k)$ defined in~(\ref{2.9}).
We rewrite (\ref{2.19}) as
\begin{eqnarray}
S_N  \!\! &=&  \!\! \sum_{\substack{(j_1,\dots,j_k),\\ 
1\le j_1\le\cdots\le j_k\le d}} \!\! \int \left(\sum_{p=1}^K
C_pc_{k_1(j_1,\dots,j_k),\dots,k_d(j_1,\dots,j_k)}
h^N_{j_1,\dots,j_k}(t_p,x_1,\dots,x_k)\right) \nonumber \\
&&\qquad\qquad\qquad\qquad \quad
Z_{G^{(N)},{j_1}}(\,dx_1)\dots Z_{G^{(N)},{j_k}}(\,dx_k) \label{2.21}
\end{eqnarray}
with
$$
h^N_{j_1,\dots,j_k}(t_p,x_1,\dots,x_k)=
f^N_{k_1(j_1,\dots,j_k),\dots,k_d(j_1,\dots,j_k)}(t_p,x_1,\dots,x_k)
$$
for all $N=1,2,\dots$, and (\ref{2.20}) as
\begin{eqnarray}
S_0&=& \sum_{\substack{(j_1,\dots,j_k),\\
1\le j_1\le \cdots\le j_k\le d}} \int\left(\sum_{p=1}^K
C_pc_{k_1(j_1,\dots,j_k),\dots,k_d(j_,\dots,j_k)} 
h^0_{j_1,\dots,j_k}(t_p,x_1,\dots,x_k)\right) \nonumber \\
&&\qquad\qquad\qquad\qquad \quad
Z_{G^{(0)},{j_1}}(\,dx_1)\dots Z_{G^{(0)},{j_k}}(\,dx_k) 
\label{2.22}
\end{eqnarray}
with
$$
h^0_{j_1,\dots,j_k}(t_p,x_1,\dots,x_k)=
f^0_{k_1(j_1,\dots,j_k),\dots,k_d(j_1,\dots,j_k)}(t_p,x_1,\dots,x_k)
$$
where the functions $k_s(j_1,\dots,j_k)$, $1\le s\le d$, are 
defined in~(\ref{2.9}).

The random integrals in formulas~(\ref{2.21}) and~(\ref{2.22})
have kernel functions of the form
\begin{eqnarray}
&&h^{N}_{j_1\dots,j_k}(x_1,\dots,x_k)
=h^{N}_{j_1\dots,j_k,t_1,\dots,t_K}(x_1,\dots,x_k) \label{2.23}\\
&&\qquad= \sum_{p=1}^K
C_pc_{k_1(j_1,\dots,j_k),\dots,k_d(j_1,\dots,j_k)}
h^N_{j_1,\dots,j_k}(t_p,x_1,\dots,x_k) 
\nonumber
\end{eqnarray}
for all $N=1,2,\dots$.
Let us introduce the measures $\mu_{N,j_1,\dots,j_k}$, $N=0,1,2,\dots$,
defined by the formula
\begin{eqnarray}
&&\mu^{(N)}_{j_1,\dots,j_k}(A)=
\mu^{(N)}_{j_1,\dots,j_k,t_1,\dots,t_K}(A) \label{2.24} \\
&&\qquad = \int_A|h_{N,j_1,\dots,j_K,t_1,\dots,t_K}(x_1,\dots,x_k)|^2  
G^{(N)}_{j_1,j_1}(\,dx_1)\dots G^{(N)}_{j_k,j_k}(\,dx_k) 
\nonumber
\end{eqnarray}
for all measurable sets $A\in{\cal B}^{k\nu}$.

We want to show with the help of Proposition~2A that the
random variables $S_N$, $N=1,2,\dots$, defined 
in~(\ref{2.21}) converge weakly to the random variable
$S_0$ defined in~(\ref{2.22}). This implies Theorem~1.3.

To prove this convergence we have to show that the functions
$h^N_{j_1,\dots,j_k}$, $n=0,1,2,\dots$, defined in (\ref{2.23}) 
and the measures $G^{(N)}_{j,j'}$, $1\le j,j'\le d$, 
$N=0,1,2,\dots$, satisfy the conditions of Proposition~2A. 
The main point is to prove relation~(\ref{2.7}) in 
condition~(b) of Proposition~2.1. To prove this we show 
that the measures $\mu^{(N)}_{j_1,\dots,j_k}$, $N=1,2,\dots$, 
defined in (\ref{2.24}) are tight, i.e. for all $\varepsilon>0$ 
there exists a $T=T(\varepsilon,j_1,\dots,j_k,t_1,\dots,t_K)$ 
such that 
$$
\mu^{N}_{j_1,\dots,j_k,t_1,\dots,t_k}
(\mathbb R^{k\nu}\setminus[-T,T]^{k\nu})<\varepsilon \quad
\textrm{for all }N=1,2,\dots.
$$

Because of the Schwarz inequality and the definition 
of the functions \hfill\break 
$h^{N}_{j_1\dots,j_k,t_1,\dots,t_K}(x_1,\dots,x_k)$
the proof of this tightness property can be reduced to the
justification of the following inequality.

Let us define for all $t=(t_1,\dots,t_\nu)\in[0,1]^\nu$,
and $N=1,2,\dots$ the measure $\mu_{N,t}$ on 
${\mathbb R}^{k\nu}$ by the formula
\begin{eqnarray*}
\mu_{N,t}(A)&=& \int_A \left|\prod_{l=1}^\nu 
\frac{\exp\left\{i\frac{]t^{(l)}N[}N(x_1^{(l)}+\cdots+x_k^{(l)})\right\}-1} 
{N\left(\exp\left\{i\frac1N(x_1^{(l)}+\cdots+x_k^{(l)})\right\}-1
\right)}\right|^2 \\
&&\qquad\qquad \qquad\qquad \qquad
G^{(N)}_{j_1,j_1}(\,dx_1)\dots G^{(N)}_{j_k,j_k}(\,dx_k) 
\end{eqnarray*}
for all $A\in{\cal B}^{k\nu}$. Then the inequality
$$
\mu_{N,t}({\mathbb R}^{k\nu}\setminus [-T,T]^{k\nu})<\varepsilon
$$
holds for all $N=1,2,\dots$, if $T\ge T_0(\varepsilon,t)$ with an
appropriate threshold index $T_0(\varepsilon,t)>0$.

I claim that the measures $\mu_{N,t}$ converge weakly to a
measure $\mu_{0,t}$ on ${\mathbb R}^{k\nu}$ as $N\to\infty$. This
convergence implies the above inequality. This convergence can
be proved similarly to Proposition~2.1. Namely we can write
\begin{eqnarray*}
&&\prod_{l=1}^\nu 
\frac{\exp\left\{i\frac{]t^{(l)}N[}N(x_1^{(l)}+\cdots+x_k^{(l)})\right\}-1} 
{N\left(\exp\left\{i\frac1N(x_1^{(l)}+\cdots+x_k^{(l)})\right\}-1
\right)} \\
&&\qquad =\frac1{N^\nu}\sum_{u\in B_N(t)}
\exp\left\{i\frac1N(u,x_1+\cdots+x_k)\right\},
\end{eqnarray*}
where $B_N(t)$ was defined in~(\ref{1.10}), and then we can 
calculate the Fourier transform of the measure $\mu_{N,t}$
in all points of the form $u=(u_1,\dots,u_k)$ with some 
$u_s=\frac {p_s}N$, $p_s\in{\mathbb Z}^\nu$, $1\le s\le k$, 
similarly to the corresponding calculation in Proposition~2.1. 
Then we can give a good asymptotic on this Fourier transform 
with the help of relation~(\ref{1.2}), and this yields the 
proof of the above mentioned convergence. Here again we apply 
a natural adaptation of the proof in Proposition~2.1. I omit 
the details of these calculations.

Thus we have proved that in our model condition~(b) of 
Proposition~2A holds. The proof of the remaining conditions is
much simpler. Similarly to the proof of Proposition~1.1 we can 
refer to Proposition~1.1 when we want to check that the
spectral measures $G^N_{j,j'}$ satisfy the demanded convergence
property. Finally, it is not difficult to check that the
functions $h^N_{j_1,\dots,j_k}$ defined in~(\ref{2.23}) satisfy
condition~A of Proposition~2A. Theorem~1.3 is proved.

\medskip
Let me finally remark that some simple and natural
modification in the proof of  Theorem~1.3 shows that this
results also holds if the random variables $S_0(t)$
in it are defined for all $t\in[0,\infty)^\nu$, (in the
way as it is explained at the end of Section~1) and not
only for $t\in[0,1]^\nu$. 

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\bibitem{2} Dobrushin, R. L. and Major, P. (1979) Non-central 
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\bibitem{3} Major, P. (2014) Multiple Wiener--It\^o Integrals. 
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\bibitem{4} Major, P. (2019) Wiener--It\^o integral 
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\end{thebibliography}

\end{document}





