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\centerline{\bf A useful lemma of Arcones}
\centerline{\it by P\'eter Major}
\centerline{Mathematical Institute of the Hungarian Academy of Sciences}

\medskip\noindent
In Lemma~1 of Arcones' paper Limit theorems for non-linear functionals
of a stationary Gaussian sequence of vectors (The Annals of Probability
1994 Vol. 22 No.~4, 2242--2274) the following result is proved (with
a slightly different notation).

\medskip\noindent
{\bf Lemma 1 in Arcones' paper.} {\it Let $X=(X_1,\dots,X_d)$ and
$Y=(Y_1,\dots,Y_d)$ be two $d$-dimensional (jointly Gaussian) standard
normal random vectors with covariances $EX_jY_k=r(j,k)$, $1\le j,k\le d$,
let $f(x_1,\dots,x_d)\in \BBB{R}^d$ be such a function for which
$Ef^2(X_1,\dots,X_d)<\infty$, and it has rank $\tau$ with some positive
integer $\tau$, i.e.
$$
Ef(X_1,\dots,X_d)P_k(X_1,\dots,X_d)=0
$$
for any polynomial $P_k(x_1,\dots,x_d)$ of order $0\le k<\tau$. Suppose that
$$
\psi=\left( \sup_{1\le j\le d}\sum_{k=1}^d|r(j,k)|\right)\vee
\left( \sup_{1\le k\le d}\sum_{j=1}^d|r(j,k)|\right)\le1. \tag1
$$
Then
$$
|Ef(X_1,\dots,X_d)f(Y_1,\dots,Y_d)|\le \psi^\tau Ef^2(X_1,\dots,X_d). \tag2
$$
}

\medskip
Here I present a more detailed proof of this result together with
some explanation about its role in the study of some probems.

\medskip\noindent
{\it Remark:}\/ First I explain the role of the above lemma in the
study of limit theorems (central or non-central limit theorems)
for non-linear functionals of vector-valued Gaussian stationary
random fields.

In the scalar valued case where limit theorems are proved for the
partial sums of random variables of the form $f(X_p)$, where $f(\cdot)$
is an appropriate real valued function on the real line, and $X_p$,
$p\in {\BBB R}^\nu$, is a stationary random field consisting of standard
normal random variables the following argument helps to simplify
the problem.

Let us expand the function $f$ with respect to the normalized
Hermite polynomials as
$f(x)=\sum\limits_{t=1}^\infty a_t\frac{H_t(x)}{\sqrt{t!}}$ with
$\sum\limits_{t=1}^\infty a_t^2<\infty$ if $Ef(X_p)=0$ and $Ef^2(X_p)<\infty$
for the random variables $X_p$ of the random field. Let $r(p,q)=EX_pX_q$,
$p,q\in{\BBB R}^\nu$, denote the covariance function of the random field.
Then $EH_t(X_p)H_{t'}(X_q)=0$ for all indices $p$ and $q$ if $t\neq t'$, and
$Ea_t\frac{H_t(X_p)}{\sqrt{t!}}a_t\frac{H_t(X_q)}{\sqrt{t!}}=a_t^2 r(p,q)^t$,
hence
$\left|Ea_t\frac{H_t(X_p)}{\sqrt{t!}}a_t\frac{H_t(X_q)}{\sqrt{t!}}\right|
\le a_t^2 |r(p,q)|^t$.

Since $Ef(X_p)f(X_q)=\sum\limits_{t=1}^\infty
Ea_t\frac{H_t(X_p)}{\sqrt{t!}}a_t\frac{H_t(X_q)}{\sqrt{t!}}$ the last
inequality implies that the high-term elements in the expansion of
the function~$f(\cdot)$ with respect to the Hermite polynomials have
a negligible contribution to the value of the expectation $Ef(X_p)f(X_q)$
if the covariance function $r(\cdot,\cdot)$ satisfies certain
conditions, (if $r(p,q)\to0$ sufficiently fast, as $|p-q|\to\infty$).
This has the consequence that in such cases they have a
negligible role in the limit theorems we are interested in.

The above lemma of Arcones is a natural generalization of the above
estimate of $Ef(X_p)f(X_q)$ to the vector-valued case. It
provides a similar reduction in the study of limit theorems for
non-linear functionals of vector-valued stationary random fields.
In such cases we consider a vector-valued Gaussian stationary
random field whose elements $X_p=(X_{p,1},\dots,X_{p,d})$, $p\in{\BBB R}^\nu$,
are $d$-dimensional random vectors with standard normal
distribution together with a function $f(x_1,\dots,x_d)$ defined
in the $d$-dimensional Euclidean space.

Let us fix two elements $X_p=(X_{p,1},\dots,X_{p,d})$
and $X_q=(X_{q,1},\dots,X_{q,d})$ of this random field, apply the
notation $X_p=(X_{p,1},\dots,X_{p,d})=X=(X_1,\dots,X_d)$,
$X_q=(X_{q,1},\dots,X_{q,d})=Y=(Y_1,\dots,Y_d)$ and
$EX_{p,j}X_{q,k}=EX_jY_k=r(j,k)$, $1\le j,k\le d$, for the elements
of the covariance matrix of these two random vectors. Arcones' lemma
gives an estimate on $|Ef(X_{p,1},\dots,X_{p,d})f(X_{q,1},\dots,X_{q,d})|$
with such a notation.

Let us remark that this approach to this problem is very similar in
the scalar and vector valued cases. Formulas~(4) and~(5) in this
note are the natural multivariate versions of the expansion of
the function $f(\cdot)$ with respect to normalized Hermite
polynomials. The estimate~(6) corresponds to the estimate 
$\left|Ea_t\frac{H_t(X_p)}{\sqrt{t!}}a_t\frac{H_t(X_q)}{\sqrt{t!}}\right|
\le a_t^2|r(p,q)|^t$ in the scalar valued case. Arcones' estimate can
be applied similarly to the corresponding result in the scalar-valued
case, only the absolute value $|r(p,q)|$ of the covariance function
has to be replaced with the function $\psi_{p,q}=\psi$ defined in
formula~(1) with the arguments $r(j,k)=r_{p,q}(j,k)=EX_{p,j}X_{q,k}$,
$1\le j,k\le d$.

\medskip\noindent
{\it The proof of Arcones' lemma.}\/ In this proof the so-called
diagram formula plays an important role. This formula deals with
the following problem. Let us have a Gaussian random vector
$X=(X_1,\dots,X_s)$ whose coordinates are standard normal random
variables, and let us calculate the expectation of the product
$H_{k_1}(X_1)\cdots H_{k_s}(X_s)$ with the help of the elements of
the covariance matrix of the random vector $X$, where $H_k(x)$
denotes the $k$-th Hermite polynomial. The diagram formula yields
a formula for this expected value with the help of some diagrams.
More explicitly, the diagram formula contains the following result.

\medskip\noindent
{\bf Diagram formula.} {\it Let $(X_1,\dots,X_s)$ be a Gaussian random
vector with $EX_1=0$, $EX_i^2=1$ for all $1\le i\le s$. Let
$r(i,j)=EX_iX_j$ for all $1\le i,j\le s$, and let $H_k(x)$ denote the
$k$-th Hermite polynomial (with leading coefficient~1). Let us take
some non-negative integers $k_1,\dots,k_s$ and consider the expected
value $EH_{k_1}(X_1)\cdots H_{k_s}(X_s)$. The diagram formula expresses
this expectation with the help of some diagrams defined in the following
way.

Let $\Gamma=\Gamma(k_1,\dots,k_s)$ denote the set of diagrams
$\gamma\in\Gamma$, defined in the following way. Each
diagram~$\gamma\in\Gamma$ consists of $s$ rows, and the $i$-th row,
(which corresponds to the random variable~$X_i$), contains $k_i$ vertices,
denoted by $(i,1),(i,2),\dots,(i,k_i)$. A diagram
$\gamma\in\Gamma$ has vertices defined in the above way, and it
contains edges with the following properties. From each vertex of
$\gamma$ starts exactly one edge. An edge can connect only vertices
from different rows. The set $\Gamma=\Gamma(k_1,\dots,k_s)$ contains
the diagrams $\gamma$ with the above properties. Given a diagram
$\gamma\in\Gamma(k_1,\dots,k_s)$ let $E(\gamma)$ denote the set of edges
contained in $\gamma$.

With the above notations the following identity holds.
$$
EH_{k_1}(X_1)\cdots H_{k_s}(X_s)
=\sum_{\gamma\in\Gamma(k_1,\dots,k_s)}
\prod_{((i,p_i),(j,p_j))\in E(\gamma)} r(i,j). \tag3
$$
In words: Given a diagram $\gamma\in\Gamma$ we attach a (possibly
negative) weight to it by giving the weight $r(i,j)$ to an edge
connecting a vertex from the $i$-th row with a vertex from the
$j$-th row, and then the weight of a diagram~$\gamma$ is defined as
the product of the weights of its edges. The expected value
we are interested in equals the sum of the weights of
the diagrams  in the set~$\Gamma(k_1,\dots,k_s)$.}

\medskip
In the proof of Arcones' lemma we exploit that the products
$\frac{H_{k_1}(X_1)}{\sqrt{k_1!}}\cdots\frac{H_{k_d}(X_d)}{\sqrt{k_d!}}$
for all possible indices $(k_1,\dots,k_d)$ yield an orthonormal
basis in the Hilbert space generated by the random variables
$X_1,\dots,X_d$, and the products
$\frac{H_{k_1}(Y_1)}{\sqrt{k_1!}}\cdots\frac{H_{k_d}(Y_d)}{\sqrt{k_d!}}$
yield an orthonormal basis in the Hilbert space generated by the
random variables $Y_1,\dots,Y_d$. It is useful to consider the
expansion of the functions $f(X_1,\dots,X_d)$ and $f(Y_1,\dots,Y_d)$
with respect to these orthonormal systems together with their following
natural decomposition.
$$
\aligned
f(X_1,\dots,X_d)&=\sum_{t=\tau}^\infty f_t(X_1,\dots,X_d) \\
f(Y_1,\dots,Y_d)&=\sum_{t=\tau}^\infty f_t(Y_1,\dots,Y_d)
\endaligned \tag4
$$
with
$$
f_t(x_1,\dots,x_d)=
\sum_{(k_1,\dots,k_d)\colon\, k_1+\cdots+k_d=t} a_{k_1,\dots,k_d}(t)
\frac{H_{k_1}(x_1)}{\sqrt{k_1!}}\cdots\frac{H_{k_d}(x_d)}{\sqrt{k_d!}} \tag5
$$
with some appropriate coefficients $a_{k_1,\dots,k_d}(t)$. Then
$\sum\limits_{(k_1,\dots,k_d)\colon\, k_1+\cdots+k_d=t} a^2_{k_1,\dots,k_d}(t)
=Ef_t^2(X_1,\dots,X_d)$.

Summation in~(4) is taken for $\tau\le t<\infty$, because the function
$f$ has rank~$\tau$. The function $f_t(\cdot)$ defined in~(5) has rank~$t$.
$EH_{k_1}(X_1)\cdots H_{k_d}(X_d) H_{k'_1}(X_1)\cdots H_{k'_d}(X_d)=0$ if
$(k_1,\dots,k_d)\neq (k_1',\dots,k_d')$, hence the random variables
$f_t(X_1,\dots,X_d)$ are orthogonal for different indices~$t$, and
$Ef^2(X_1,\dots,X_d)=\sum\limits_{t=\tau}^\infty Ef^2_t(X_1,\dots,X_d)$.

Let us also observe that $Ef_t(X_1,\dots,X_d) f_{t'}(Y_1,\dots,Y_d)=0$
if $t\neq t'$. This follows from the fact that
$EH_{k_1}(X_1)\cdots H_{k_d}(X_d)H_{k'_1}(Y_1)\cdots H_{k'_d}(Y_d)=0$
if $t\neq t'$ for $t=k_1+\cdots+k_d$ and  $t'=k'_1+\cdots+k'_d$. The
last identity follows from the diagram formula, because both the
random variables $X_1,\dots,X_d$ and  the random variables
$Y_1,\dots,Y_d$ are orthogonal. These orthogonalities imply that
only such diagrams yield a non-zero contribution to the expectation
of the above product of Hermite polynomials of standard normal
random variables in the application of the diagram formula in which
each edge connects a vertex from a row corresponding to a random
variable $X_i$ with a vertex from a row corresponding to a random
variable $Y_j$. But there are no such diagrams if $t\neq t'$.

\medskip\noindent
{\it Remark.}\/  Given a set of Gaussian random variables $X_t$,
$t\in T$, and the Hilbert space $\Cal H$ generated by these random
variables, there is a so-called Fock--space representation of this
Hilbert space $\Cal H$. In the above discussion we applied this
Fock--space representation in a special case, and showed some of
its basic properties.

\medskip
I claim that under the conditions of Lemma~1 in Arcones' paper the
following inequality holds for all $t\ge\tau$:
$$
|Ef_t(X_1,\dots,X_d)f_t(Y_1,\dots,Y_d)|\le \psi^t Ef_t^2(X_1,\dots,X_d).
\tag6
$$

It is not difficult to see that relation~(6) implies Arcones' lemma.
Indeed, it yields (together with the condition $\psi\le1$) that
$$
\align
&|Ef(X_1,\dots,X_d)f(Y_1,\dots,Y_d)|
\le\sum_{t=\tau}^\infty |Ef_t(X_1,\dots,X_d)f_t(Y_1,\dots,Y_d)| \\
&\qquad \le\sum_{t=\tau}^\infty \psi^t Ef_t^2(X_1,\dots,X_d)
\le\psi^\tau\sum_{t=\tau}^\infty Ef_t^2(X_1,\dots,X_d)
=\psi^\tau Ef^2(X_1,\dots,X_d).
\endalign
$$
Formula~(6) will be proved with the help of the following lemma.

\medskip\noindent
{\bf Lemma.} {\it Let two random vectors $(X_1,\dots,X_d)$,
$(Y_1,\dots,Y_d)$ and a function $f$ of $d$ variables satisfy
the conditions of Lemma~1 in Arcones' paper. Let us fix an integer
$t\ge\tau$ and a vector $(l^{(0)}_1,\dots,l^{(0)}_d)$ or
$(m^{(0)}_1,\dots,m^{(0)}_d)$ with non-negative integer coordinates
such that $l^{(0)}_1+\cdots+l^{(0)}_d=t$, and $m^{(0)}_1+\cdots+m^{(0)}_d=t$.
Then the following inequalities hold:
$$
\aligned
&\sum_{(m_1,\dots,m_d)\colon\, m_1+\cdots+m_d=t}
\left| EH_{l^{(0)}_1}(X_1)\cdots H_{l^{0)}_d}(X_d)
\frac {H_{m_1}(Y_1)\cdots H_{m_d}(Y_d)}{m_1!\cdots m_d!}\right| \\
&\qquad\qquad \le\prod_{i=1}^d (|r(i,1)|+\cdots+ |r(i,d)|)^{l^{(0)}_i}
\le\psi^t \\
&\sum_{(l_1,\dots,l_d)\colon\, l_1+\cdots+l_d=t}
\left| EH_{m^{(0)}_1}(Y_1)\cdots H_{m^{(0)}_d}(Y_d)
\frac {H_{l_1}(X_1)\cdots H_{l_d}(X_d)}{l_1!\cdots l_d!}\right| \\
&\qquad\qquad \le\prod_{j=1}^d (|r(1,j)|+\cdots+|r(d,j|)^{m^{(0)}_j}
\le\psi^t.
\endaligned \tag7
$$
}

\medskip
First I prove formula~(6) with the help of the Lemma, and then I prove
the Lemma.

To prove formula~(6) write the inequality
$$
\align
&|Ef_t(X_1,\dots,X_d)f_t(Y_1,\dots,Y_d)| \\
&\qquad \le\sum_{(l_1,\dots,l_d)\colon\,l_1+\cdots+l_d=t} 
\sum_{(m_1,\dots,m_d)\colon\,m_1+\cdots+m_d =t}
\left|\frac{a_{l_1,\dots,l_d}(t)}{\sqrt{l_1!\cdots l_d!}}\cdot
\frac{a_{m_1,\dots,m_d}(t)}{\sqrt{m_1!\cdots m_d!}}\right| \\
&\qquad\qquad\qquad |E H_{l_1}(X_1)\cdots H_{l_d}(X_d)
H_{m_1}(Y_1)\cdots H_{m_d}(Y_d)|.
\endalign
$$

By applying this inequality together with the relation
$$
\left|\frac{a_{l_1,\dots,l_d}(t)}{\sqrt{l_1!\cdots l_d!}}\cdot
\frac{a_{m_1,\dots,m_d}(t)}{\sqrt{m_1!\cdots m_d!}}\right|
\le\frac12\left(\frac{a^2_{l_1,\dots,l_d}(t)}{m_1!\cdots m_d!}+
\frac{a^2_{m_1,\dots,m_d}(t)}{l_1!\cdots l_d!}\right)
$$
we get with the help of the Lemma that
$$
\align
&|Ef_t(X_1,\dots,X_d)f_t(Y_1,\dots,Y_d)| 
\le\sum_{(l_1,\dots,l_d)\colon\,l_1+\cdots+l_d=t} \frac{a^2_{l_1\,\dots,l_d}(t)}2 \\
&\qquad\qquad \sum_{(m_1,\dots,m_d)\colon\,m_1+\cdots+m_d =t}
\left| EH_{l_1}(X_1)\cdots H_{l_d}(X_d)
\frac {H_{m_1}(Y_1)\cdots H_{m_d}(Y_d)}{m_1!\cdots m_d!}\right| \\
&\qquad\qquad + \sum_{(m_1,\dots,m_d)\colon\,m_1+\cdots+m_d=t}   
\frac{a^2_{m_1,\dots,m_d}(t)}2 \\
&\qquad\qquad\sum_{(l_1,\dots,l_d)\colon\,l_1+\cdots+l_d =t}
\left| EH_{m_1}(Y_1)\cdots H_{m_d}(Y_d)
\frac {H_{l_1}(X_1)\cdots H_{l_d}(X_d)}{l_1!\cdots l_d!}\right| \\
&\qquad \le \psi^t \sum_{(l_1,\dots,l_d)\colon\,l_1+\cdots+l_d=t} 
a^2_{l_1,\dots,l_d}(t)=\psi^t Ef_t^2(X_1,\dots,X_d),
\endalign
$$
i.e. relation~(6) holds true.

\medskip
Next I prove an identity which is actually not applied in the
proof of the Lemma. But relation~(7) can be proved by a small
modification of this proof which may explain the idea behind 
the Lemma. To formulate this identity I fix a vector
$(l^{(0)}_1,\dots,l^{(0)}_d)$ with non-negative integer coordinates
such that $l^{(0)}_1+\cdots+l^{(0)}_d=t$. I formulate the following
identity with its help.
$$
\aligned
&\sum_{(m_1,\dots,m_d)\colon\, m_1+\cdots+m_d=t}
EH_{l^{(0)}_1}(X_1)\cdots H_{l^{0)}_d}(X_d)
\frac {H_{m_1}(Y_1)\cdots H_{m_d}(Y_d)}{m_1!\cdots m_d!} \\
&\qquad\qquad
=\prod_{i=1}^d (r(i,1)+\cdots+r(i,d))^{l^{(0)}_i}.
\endaligned \tag8
$$

In the proof of~(8) all terms in the sum at the left-hand side of
this formula will be calculated with the help of the diagram
formula. In this proof the numbers $l^{(0)}_i$, $m_j$, $u(\cdot,\cdot)$
will always denote non-negative integers.

Let us fix some vector $(m_1,\dots,m_d)$ such that $m_1+\cdots+m_d=t$,
and let us compute the expected value
$$
EH_{l^{(0)}_1}(X_1)\cdots H_{l^{0)}_d}(X_d)
\frac {H_{m_1}(Y_1)\cdots H_{m_d}(Y_d)}{m_1!\cdots m_d!} \tag9
$$
by means of the diagram formula.

In this calculation only such diagrams
$\gamma\in\Gamma(l^{(0)}_1,\dots,l^{(0)}_d,m_1,\dots,m_d)$ have to be
considered in which all edges connect a vertex from a row indexed
by $i$ with some $1\le i\le d$ with a vertex from a row indexed by
$d+j$ with some $1\le j\le d$, i.e. a vertex from a row
corresponding to some $X_i$ with a vertex in a row corresponding to
some $Y_j$. The omitted diagrams yield zero contribution in the
diagram formula.

Let us define a partition of these diagrams in the following way.
Define the set $\Cal{U}(l^{(0)}_1,\dots,l^{(0)}_d,m_1,\dots,m_d)$
consisting of some vectors by the formula
$$
\align
&\Cal {U}(l^{(0)}_1,\dots,l^{(0)}_d,m_1,\dots,m_d)
=\biggl\{\{ u(i,j),\; 1\le i,j\le d\}\colon\; \\
&\qquad
\sum_{j=1}^d u(i,j)=l^{(0)}_i \text{ for all } 1\le i\le d,
\quad \sum_{i=1}^d u(i,j)=m_j \text{ for all } 1\le j\le d \biggr\},
\endalign
$$
and introduce the set of diagrams
$$
\align
&\Gamma(\{ u(i,j),\; 1\le i,j\le d\})=\{\gamma\colon\;
\gamma\in\Gamma(l^{(0)}_1,\dots,l^{(0)}_d,m_1,\dots,m_d), \\
&\qquad
\text{ there are $u(i,j)$ vertices between the $i$-th and $d+j$-th rows of }
\gamma, \; 1\le i,j\le d\}
\endalign
$$
for all $\{ u(i,j),\; 1\le i,j\le d\}\in
\Cal {U}(l^{(0)}_1,\dots,l^{(0)}_d,m_1,\dots,m_d)$.

\medskip
Let us consider the sets 
$\Gamma(\{u(i,j),\; 1\le i,j\le d\})$ with all possible arguments 
\hfill\break
$\{ u(i,j),\; 1\le i,j\le d\}\in
\Cal {U}(l^{(0)}_1,\dots,l^{(0)}_d,m_1,\dots,m_d)$. They give a partition
of those diagrams of 
$\Gamma(l^{(0)}_1,\dots,l^{(0)}_d,m_1,\dots,m_d)$ that have to be taken
into account when the expression in~(9) is computed by means of
the diagram formula. For each argument
$\{ u(i,j),\; 1\le i,j\le d\}$ the set
$\Gamma(\{ u(i,j),\; 1\le i,j\le d\})$ contains
$$
\frac{\prod\limits_{i=1}^d l^{(0)}_i!}
{\prod\limits_{1\le i,j\le d}u(i,j)!} \cdot
\frac{\prod\limits_{j=1}^d m_j!}
{\prod\limits_{1\le i,j\le d}u(i,j)!}
\prod\limits_{1\le i,j\le d} u(i,j)!
=\frac{\prod\limits_{i=1}^d l^{(0)}_i!\prod\limits_{j=1}^d m_j!}
{\prod\limits_{1\le i,j\le d} u(i,j)!}
$$
diagrams, and all these diagrams have weight
$\prod\limits_{1\le i,j\le d} r(i,j)^{u(i,j)}$. Hence in the calculation
of the expression in~(9) by means of the diagram formula the sum of
the contributions of the weights of the diagrams
$\gamma\in\Gamma(\{u(i,j),\; 1\le i,j\le d\})$ equals
$$
\frac{\prod\limits_{i=1}^d l^{(0)!}_i\prod\limits_{1\le i,j\le d} r(i,j)^{u(i,j)}}
{\prod\limits_{1\le i,j\le d} u(i,j)!}
=\prod_{i=1}^d\left( l^{(0)}_i!
\left(\prod_{j=1}^d \frac{r(i,j)^{u(i,j)}}{u(i,j)!}\right)\right) .
$$
The sum of these expressions for all 
$\{u(i,j),\,1\le i,j\le d\}\in\Cal{U}(l^{(0)}_1,\dots,l^{(0)}_d,m_1,\dots,m_d)$
equals the expression in formula~(9). In such a way we get the following
identity:
$$
\align
&EH_{l^{(0)}_1}(X_1)\cdots H_{l^{0)}_d}(X_d)
\frac {H_{m_1}(Y_1)\cdots H_{m_d}(Y_d)}{m_1!\cdots m_d!} \\
&\qquad =\sum \Sb \{(u(i,j),\; 1\le i,j\le d)\colon \\
\sum_{j=1}^d u(i,j)=l^{(0)}_i, \; 1\le i\le d,\\        
\sum_{j=1}^d u(i,j)=m_j, \; 1\le j\le d\}\endSb       
\prod_{i=1}^d\left( l^{(0)}_{i}!
\left(\prod_{j=1}^d \frac{r(i,j)^{u(i,j)}}{u(i,j)!}\right)\right). \tag10
\endalign
$$
By summing up these identities for all vectors $(m_1,\dots,m_d)$ such
that $m_1+\cdots+m_d=t$ we get that
$$
\align
&\sum_{(m_1,\dots,m_d)\colon\, m_1+\cdots+m_d=t}
EH_{l^{(0)}_1}(X_1)\cdots H_{l^{0)}_d}(X_d)
\frac {H_{m_1}(Y_1)\cdots H_{m_d}(Y_d)}{m_1!\cdots m_d!} \\
&\qquad =\sum \Sb\{ (u(i,j),\; 1\le i,j\le d)\colon \\
\sum_{j=1}^d u(i,j)=l^{(0)}_i, \; 1\le i\le d\}\endSb
\prod_{i=1}^d\left( l^{(0)}_{i}!
\left(\prod_{j=1}^d \frac{r(i,j)^{u(i,j)}}{u(i,j)!}\right)\right). \tag11
\endalign
$$

To prove identity~(8) I present a new  identity in the next
formula~(12). In the second line of this formula the expression at
the right-hand side of formula~(11) is rewritten as the product of
certain expressions. Then this product is rewritten in a simpler
form by means of the multinomial theorem. Here is formula~(12).
$$
\align
&\sum \Sb\{ (u(i,j),\; 1\le i,j\le d)\colon \\
\sum_{j=1}^d u(i,j)=l^{(0)}_i, \; 1\le i\le d\}\endSb
\prod_{i=1}^d\left( l^{(0)}_{i}!\left(\prod_{j=1}^d
\frac{r(i,j)^{u(i,j)}}{u(i,j)!}\right)\right) \\
&\qquad=\prod_{i =1}^d\left(
\sum\Sb \{(u(i,j),\;1\le j\le d)\colon  \\
\sum_{j=1}^d u(i,j)=l^{(0)}_i\} \endSb
l^{(0)}_{i}!\left(\frac{r(i,1)^{u(i,1)}}{u(i,1)!}
\cdots\frac{r(i,d)^{u(i,d)}}{u(i,d)!}\right)\right)\\
&\qquad=\prod_{i=1}^d\left(r(i,1)+\cdots+r(i,d)\right)^{l^{(0)}_i}. \tag12
\endalign
$$
Formulas~(11) and~(12) together imply identity~(8).

To prove the first identity in formula~(12) let us first observe
that if we carry out the term by term multiplications in the product
of sums at the right-hand side of this identity, then both sides
of this identity are sums of terms which have the following form:
$$
\prod_{i =1}^d l^{(0)}_{i}!\left(\frac{r(i,1)^{u(i,1)}}{u(i,1)!}
\cdots\frac{r(i,d)^{u(i,d)}}{u(i,d)!}\right).
$$
We have to understand that the same terms appear in the sums
of the two sides of this identity. But in both cases those terms
of the above form appear in these sums which are defined by
those vectors $(u(i,1),\dots, u(i,d))$, $1\le i\le d$, which
satisfy the identity $u(i,1)+\dots+u(i,d)=l^{(0)}_i$ for all
$1\le i\le d$. Hence the first identity in~(12) holds.

The second identity in formula~(12) is a consequence of 
the multinomial theorem. 

\medskip
The proof of the first inequality in the first statement of formula~(7)
can be obtained with some small, natural modifications in the proof
of relation~(8).

Let us first observe that the absolute value of the expression at the
left-hand side of~(10) is smaller than the expression we get by
replacing the terms $r(i,j)$ by $|r(i,j)|$ at the right-hand side of
this identity. Then summing up these inequalities we get such a
version of relation~(11) where in the sum at the left-hand side the
summands are replaced by their absolute value, at the right-hand side
the variables $r(i,j)$ are replaced by their absolute value $|r(i,j)|$,
and in this  new version of formula~(11) the expression at the left-hand
side is less than or equal to the expression at the right-hand side.

Formula~(12) remains valid if we replace $r(i,j)$ by $|r(i,j)|$
in it. These new versions of formulas~(11) and~(12) imply
the first inequality in the first statement of~(7).

\medskip
The proof of the second inequality in the first statement of~(7)
is much simpler.

Since $|r(i,1)|+\cdots+|r(i,d)|\le\psi$ for all $1\le i\le d$, and
$l^{(0)}_1+\cdots+l^{(0)}_d=t$ 
$$
\prod_{i=1}^d (|r(i,1)|+\cdots+ |r(i,d)|)^{l^{(0)}_i}
\le\prod_{i=1}^d \psi^{l^{(0)}_i}=\psi^t.
$$

The second statement of~(7) is equivalent to the first one. Only the
variables $l^{(0)}_i$ and $m_j$ must be replaced by the variables $m^{(0)}_j$
and $l_i$, $1\le i,j\le d$, and the role of the random vectors
$(X_1,\dots,X_d)$ and $(Y_1,\dots,Y_d)$ have to be exchanged. Lemma~1 in
Arcones' paper is proved.


\bye



