\beginsection Appendix A.

{\it The proof of some results about Vapnik--\v{C}ervonenkis classes.}

\medskip\noindent
{\it Proof of Theorem 5.1. (Sauer's lemma).}\/ This result has several
different proofs. Here I write down a relatively simple proof
of P. Frankl and J. Pach which appeared in~[15]. It is based on some
linear algebraic arguments.

The following equivalent reformulation of Sauer's lemma will be
proved. Let us take a set $S=S(n)$ consisting of $n$ elements and
a class $\Cal E$ of subsets of $S$ consisting of $m$ elements
$E_1,\dots,E_m\subset S$. Assume that $m\ge m_0+1$ with
$m_0=m_0(n,k)=\binom n0+\binom n1+\cdots+\binom n{k-1}$. Then there
exists a set $F\subset S$ of cardinality $k$ which the class
of sets $\Cal E$ shatters. Actually, it is enough to show
that there exists a set $F$ of cardinality greater than or equal
to~$k$ which the class of sets $\Cal E$ shatters, because if a set
has this property, then all of its subsets have it. This latter
statement will be proved.

To prove this statement let us first list the subsets
$X_0,\dots,X_{m_0}$ of the set $S$ of cardinality less than or equal
to $k-1$, and correspond to all sets $E_i\in\Cal E$ the vector
$e_i=(e_{i,1},\dots,e_{i,m_0})$, $1\le i\le m$, with elements
$$
e_{i,j}=\left\{\aligned 1&\quad\text{if }X_j\subseteq E_i \\
                        0&\quad\text{if }X_j\not\subseteq E_i
\endaligned \right.     \qquad 1\le i\le m, \text{ and } 1\le j\le m_0.
$$

Since $m>m_0$, the vectors $e_1,\dots,e_m$ are linearly dependent.
Because of the definition of the vectors $e_i$, $1\le i\le m$, this can
be expressed in the following way: There is a non-zero vector
$(f(E_1),\dots,f(E_m))$ such that
$$
\sum_{E_i\colon\; E_i\supseteq X_j} f(E_i)=0 \quad \text{for all }
1\le j\le m_0.  \tag A1
$$

Let $F$, $F\subset S$, be a {\it minimal}\/ set with the property
$$
\sum_{E_i\colon\; E_i\supseteq F} f(E_i)=\alpha\neq0. \tag A2
$$
Such a set $F$ really exists, since every maximal element of the
family $\{E_i\colon\; 1\le i\le m,\, f(E_i)\neq0\}$ satisfies
relation (A2). The requirement that $F$ should be a minimal set means
that if $F$ is replaced by some $H\subset F$, $H\neq F$, at the
left-hand side of~(A2), then this expression equals zero. The inequality
$|F|\ge k$ holds because of relation (A1) and the definition of the
sets $X_j$.

Introduce the quantities
$$
Z_F(H)=\sum_{E_i\colon\; E_i\cap F=H} f(E_i)
$$
for all $H\subseteq F$.

Then $Z_F(F)=\alpha$, and for any set of the form $H=F\setminus\{x\}$,
$x\in F$,
$$
Z_F(H)=\sum_{E_i\colon\; E_i\cap F=H} f(E_i)
=\sum_{E_i\colon\; E_i\supseteq H}f(E_i)
-\sum_{E_i\colon\; E_i\supseteq F}f(E_i)=0-\alpha=-\alpha
$$
because of the minimality property of the set $F$.

Moreover, the identity
$$
Z_F(H)=(-1)^p\alpha \quad\text{for all } H\subseteq F \text{ such that }
|H|=|F|-p, \; 0\le p\le |F|. \tag A3
$$
holds. To show relation (A3) observe that
$$
Z_F(H)=\sum_{E_i\colon\; E_i\cap F=H} f(E_i)=\sum_{j=0}^p
(-1)^j\sum_{G\colon\;H\subset G\subset F,\;|G|=|H|+j}
\sum_{E_i\colon\; E_i\supseteq G}f(E_i) \tag A4
$$
for all sets $H\subset F$ with cardinality $|H|=|F|-p$. Identity~(A4)
holds, since the term $f(E_i)$ is counted at the right-hand side of~(A4)
$\summ_{j=0}^l (-1)^j\binom lj=(1-1)^l=0$ times if $E_i\cap F=G$ with
some $H\subset G\subseteq F$ with $|G|=|H|+l$ elements, $1\le l\le p$,
while in the case $E_i\cap F=H$ it is counted once. Relation (A4)
together with (A2) and the minimality property of the set~$F$ imply
relation~(A3).

It follows from relation (A3) and the definition of the function
$Z_F(H)$ that for all sets $H\subseteq F$ there exists some set $E_i$
such that $H=E_i\cap F$, i.e. $F$ is shattered by $\Cal E$. Since
$|F|\ge k$, this implies Theorem~5.1.

\medskip\noindent
{\it Proof of Theorem 5.3.}\/ Let us fix an arbitrary set
$F=\{x_1,\dots,x_{k+1}\}$ of the set $X$, and consider the set of
vectors $\Cal G_k(F)=\{(g(x_1),\dots,g(x_{k+1}))\colon\; g\in\Cal G_k\}$
of the $k+1$-dimensional space $R^{k+1}$. By the conditions of
Theorem~5.3 $\Cal G_k(F)$ is an at most $k$-dimensional subspace of
$R^{k+1}$. Hence there exists a non-zero vector $a=(a_1,\dots,a_{k+1})$
such that $\summ_{j=1}^{k+1} a_jg(x_j)=0$ for all $g\in\Cal G_k$. We
may assume that the set $A=A(a)=\{j\colon\; a_j<0,\, 1\le j\le k+1\}$ 
is non-empty, by multiplying the vector $a$ by $-1$ if it is necessary.

Thus the identity
$$
\sum_{j\in A} a_jg(x_j)=\sum_{j\in \{1,\dots,k+1\}\setminus A}
(-a_j)g(x_j),\qquad \text{for all }g\in\Cal G_k \tag A5
$$
holds. Put $B=\{x_j\colon\; j\in A\}$. Then $B\subset F$, and
$F\setminus B\neq\{x\colon\; g(x)\ge0\}\cap F$ for all $g\in\Cal G_k$.
Indeed, if there were some $g\in \Cal G_k$ such that
$F\setminus B=\{x\colon\; g(x)\ge0\}\cap F$, then the left-hand side
of the equation (A5) would be strictly positive (as $a_j<0$, $g(x_j)<0$
if $j\in A$, and $A\neq\emptyset$) its right-hand
side would be non-positive for this $g\in\Cal G_k$, and this is a
contradiction.

The above proved property means that $\Cal D$ shatters no set
$F\subset X$ of cardinality~$k+1$. Hence Theorem~5.1 implies that
$\Cal D$ is a Vapnik--\v{C}ervonenkis class.

\beginsection Appendix B. The proof of the diagram formula for
Wiener--It\^o integrals.

The proof of Theorem (10.2A) (the diagram formula for the product of
two Wiener--It\^o integrals) will be started with the proof of
inequality (10.11). To show that this relation holds let us observe
that the Cauchy inequality yields the following bound on the function
$F_\gamma$ defined in~(10.10) (with the notation introduced there):
$$
\aligned
&F^2_\gamma(x_{(1,j)},x_{(2,j')},\,\;(1,j)\in
V_1(\gamma),\, (2,j')\in V_2(\gamma))\\
&\qquad\le\int f^2(x_{\alpha_\gamma(1,1)},\dots,x_{\alpha_\gamma(1,k)})
\prod_{(2,j)\in\{(2,1),\dots,(2,l)\}\setminus
V_2(\gamma)} \mu(\,dx_{(2,j)})\\
&\qquad\qquad
\int g^2(x_{(2,1)},\dots,x_{(2,l)})
\prod_{(2,j)\in\{(2,1),\dots,(2,l)\}\setminus
V_2(\gamma)}\mu(\,dx_{(2,j)}).
\endaligned \tag B1
$$
The expression at the right-hand side of inequality~(B1) is the
product of two functions with different arguments. The first
function has arguments $x_{(1,j)}$ with $(1,j)\in V_1(\gamma)$ and
the second one $x_{(2,j')}$ with $(2,j')\in V_2(\gamma)$.
By integrating both sides of inequality~(B1) with respect to these 
arguments we get inequality~(10.11).

Relation (10.12) will be proved first for the product of the
Wiener--It\^o integrals of two elementary functions. Let us consider
two (elementary) functions $f(x_1,\dots,x_k)$ and $g(x_1,\dots,x_l)$
given in the following form: Let some disjoint sets $A_1,\dots,A_M$,
$\mu(A_s)<\infty$, $1\le s\le M$, be given together with some real
numbers $c(s_1,\dots,s_k)$ indexed with such $k$-tuples
$(s_1,\dots,s_k)$, $1\le s_j\le M$, $1\le j\le k$, for which the
numbers $s_1,\dots,s_k$ in a $k$-tuple are all different. Put
$f(x_1,\dots,x_k)=c(s_1,\dots,s_k)$ on the rectangles
$A_{s_1}\times\cdots\times A_{s_k}$ with edges $A_s$, indexed
with the above $k$-tuples, and let $f(x_1,\dots,x_k)=0$ outside of
these rectangles. Take similarly some disjoint sets
$B_1,\dots,B_{M'}$, $\mu(B_t)<\infty$, $1\le t\le M'$, and some
real numbers $d(t_1,\dots,t_l)$, indexed with such $l$-tuples
$(t_1,\dots,t_l)$, $1\le t_{j'}\le M'$, $1\le j'\le l$, for which
the numbers $t_1,\dots,t_l$ in an $l$-tuple are different. Put
$g(x_1,\dots,x_l)=d(t_1,\dots,t_l)$ on the rectangles
$B_{t_1}\times\cdots\times B_{t_l}$ with edges
indexed with the above introduced $l$-tuples, and let
$g(x_1,\dots,x_l)=0$ outside of these rectangles.

Let us take some small number $\e>0$ and rewrite the above
introduced functions $f(x_1,\dots,x_k)$ and $g(x_1,\dots,x_l)$
with the help of this number $\e>0$ in the following way. Divide
the sets $A_1,\dots,A_M$ to smaller sets
$A_1^\e,\dots,A_{M(\e)}^\e$, $\bigcupp_{s=1}^{M(\e)} A_s^\e=
\bigcupp_{s=1}^{M} A_s$, in such a way that all sets
$A_1^\e,\dots,A_{M(\e)}^\e$ are disjoint, and $\mu(A^\e_s)\le\e$,
$1\le s\le M(\e)$. Similarly, take sets
$B_1^\e,\dots,B_{M'(\e)}^\e$, $\bigcupp_{t=1}^{M'(\e)} B_t^\e
=\bigcupp_{t=1}^{M'} B_t$, in such a way that all sets
$B_1^\e,\dots,B_{M'(\e)}^\e$ are disjoint, and $\mu(B^\e_t)\le\e$,
$1\le t\le M'(\e)$. Beside this, let us also demand that two sets
$A_s^\e$ and $B_t^\e$, $1\le s\le M(\e)$, $1\le t\le M'(\e)$,
are either disjoint or they agree. Such a partition exists because
of the non-atomic property of measure $\mu$. The above defined 
functions $f(x_1,\dots,x_k)$ and $g(x_1,\dots,x_l)$ can be 
rewritten by means of these new sets $A^\e_s$ and $B^\e_t$. 
Namely, let $f(x_1,\dots,x_k)=c^\e(s_1,\dots,s_k)$ on the rectangles
$A^\e_{s_1}\times\cdots\times A^\e_{s_k}$ with $1\le s_j\le M(\e)$,
$1\le j\le k$, with different indices $s_1,\dots,s_k$, where
$c^\e(s_1,\dots,s_k)=c(p_1,\dots,p_k)$ with those indices
$(p_1,\dots,p_k)$ for which
$A^\e_{s_1}\times\cdots\times A^\e_{s_k}\subset
A_{p_1}\times\cdots\times A_{p_k}$. The function $f$ disappears
outside of these rectangles. The function $g(x_1,\dots,x_l)$ can be
written similarly in the form $g(x_1,\dots,x_l)
=d^\e(t_1,\dots,t_l)$ on the rectangles
$B^\e_{t_1}\times\cdots\times B^\e_{t_l}$ with
$1\le t_{j'}\le M'(\e)$, $1\le j'\le l$, and different indices,
$t_1,\dots,t_l$. Beside this, the function $g$ disappears outside
of these rectangles.

The above representation of the functions $f$ and $g$ through a
parameter $\e$ is useful, since it enables us to give a good
asymptotic formula for the product $k!Z_{\mu,k}(f)l!Z_{\mu,l}(g)$
which yields the diagram formula for the product of Wiener--It\^o
integrals of elementary functions with the help of a limiting
procedure $\e\to0$.

Fix a small number $\e>0$, take the representation of the functions
$f$ and $g$ with its help, and write
$$
k!Z_{\mu,k}(f)l!Z_{\mu,l}(g)=\sum_{\gamma\in \Gamma(k,l)} Z_\gamma(\e)
\tag B2
$$
with
$$
Z_\gamma(\e)={\sum}^\gamma c^\e(s_1,\dots,s_k)
d^\e(t_1,\dots,t_l)
\mu_W(A^\e_{s_1})\dots\mu_W(A^\e_{s_k})
\mu_W(B^\e_{t_1})\dots\mu_W(B^\e_{t_l}),  \tag B3
$$
where $\Gamma(k,l)$ denotes the class of diagrams introduced before
the formulation of Theorem~10.2A, and $\sum^\gamma$ denotes
summation for such $k+l$-tuples $(s_1,\dots,s_k,t_1,\dots,t_l)$,
$1\le s_j\le M(\e)$, $1\le j\le k$, and $1\le t_{j'}\le M'(\e)$,
$1\le j'\le l$, for which $A^\e_{s_j}=B^\e_{t_{j'}}$ if
$((1,j),(2,j'))\in E(\gamma)$, i.e.\ if it is an edge of $\gamma$, 
and otherwise all sets $A^\e_{s_j}$ and $B^\e_{t_{j'}}$ are 
disjoint. (This sum also depends on $\e$.) In the case of an empty 
sum $Z_\gamma(\e)$ equals zero.

For all $\gamma\in\Gamma(k,l)$ the expression $Z_\gamma$ will be
written in the form
$$
Z_\gamma(\e)=Z_\gamma^{(1)}(\e)+Z_\gamma^{(2)}(\e),
\quad \gamma\in\Gamma(k,l),   \tag B4
$$
with
$$
\aligned
Z^{(1)}_\gamma(\e)&={\sum}^\gamma c^\e(s_1,\dots,s_k)
d^\e(t_1,\dots,t_l)\\
&\qquad\prod_{j\colon\; (1,j)\in V_1(\gamma)}\mu_W(A^\e_{s_j})
\prod_{j'\colon\; (2,j')\in V_2(\gamma)} \mu_W(B^\e_{t_{j'}})\\
&\qquad \prod_{j\colon\; (1,j)\in\{(1,1),\dots,(1,k)\}
\setminus V_1(\gamma)}\mu(A^\e_{s_j})
\endaligned \tag B5
$$
and
$$ \allowdisplaybreaks
\align
Z^{(2)}_\gamma(\e)&={\sum}^\gamma
c^\e(s_1,\dots,s_k) d^\e(t_1,\dots,t_l)\\
&\qquad \prod_{j\colon\; (1,j)\in V_1(\gamma)}\mu_W(A^\e_{s_j})
\prod_{j'\colon\; (2,j')\in V_2(\gamma)} \mu_W(B^\e_{t_{j'}}) \\
&\qquad \biggl[\prod_{j\colon\; (1,j)\in
\{(1,1),\dots,(1,k)\}\setminus V_1(\gamma)}\mu_W(A^\e_{s_j})
\prod_{j'\colon\; (2,j')\in \{(2,1),\dots,(2,l)\}\setminus
\in V_2(\gamma)}\mu_W(B^\e_{t_{j'}}) \\
&\qquad\qquad -\prod_{j\colon\; (1,j)\in\{(1,1),\dots,(1,k)\}
\setminus V_1(\gamma)}\mu(A^\e_{s_j})\biggr], \tag B6
\endalign
$$
where $V_1(\gamma)$ and $V_2(\gamma)$ (introduced before formula~(10.9)
during the preparation to the formulation of Theorem~10.2A) are the 
sets of vertices in the first and second row of the diagram $\gamma$ 
from which no edge starts.

I claim that there is some constant $C>0$ not depending on $\e$
such that
$$
E\(|\gamma|!Z_{\mu,|\gamma|}(F_\gamma)-
Z^{(1)}_\gamma(\e)\)^2\le C\e \quad \text{for all }
\gamma\in\Gamma(k,l) \tag B7
$$
with the Wiener--It\^o integral with the kernel function $F_\gamma$ 
defined in (10.9), (10.9a) and (10.10), and
$$
E\(Z^{(2)}_\gamma(\e)\)^2\le C\e\quad\text{for all }
\gamma\in\Gamma(k,l). \tag B8
$$

Relations (B7) and (B8) imply relation (10.12)
if $f$ and $g$ are elementary functions. Indeed, they imply that
$$
\lim_{\e\to0}\left\|\,|\gamma|!Z_{\mu,|\gamma|}(F_\gamma)-
Z_\gamma(\e)\right\|_2\to0 \quad\text{for all }\gamma\in\Gamma(k,l),
$$
and this relation together with (B2) yield relation (10.12) with
the help of a limiting procedure $\e\to0$.

To prove relation (B7) let us introduce the function
$$
\align
&F^\e_\gamma(x_{(1,j)},x_{(2,j')},\; (1,j)\in
V_1(\gamma),\; (2,j')\in  V_2(\gamma))\\
&\qquad=
F_\gamma(x_{(1,j)},x_{(2,j')},\; (1,j)\in V_1(\gamma),\;
(2,j')\in V_2(\gamma))\\
&\qquad\qquad\text{if } x_{(1,j)}\in A^\e_{s_j},\text{ for all }
(1,j)\in V_1(\gamma),\\
&\qquad\qquad\text{ } x_{(2,j')}\in B^\e_{t_{j'}},
\text{ for all } (2,j')\in V_2(\gamma)), \quad\text{and}\\
& \qquad\qquad\text{ all sets } A^\e_{s_j},\; (1,j)\in V_1(\gamma), 
\text{ and } B^\e_{t_{j'}}, \; (2,j')\in V_2(\gamma) 
\text{ are different.} 
\endalign
$$
with the function~$F_\gamma$ defined in~(10.9a) and~(10.10), and 
put
$$
F^\e_\gamma(x_{(1,j)},x_{(2,j')},\; (1,j)\in V_1(\gamma),\;
(2,j')\in V_2(\gamma))=0 \quad
\text{otherwise.}
$$

The function $F_\gamma^\e$ is elementary, and a comparison of its
definition with relation~(B5) and the definition of the function
$F_\gamma$ yield that
$$
Z_\gamma^{(1)}(\e)=|\gamma|! Z_{\mu,|\gamma|}(F_\gamma^\e). \tag B9
$$
The function $F^\e_\gamma$ slightly differs from $F_\gamma$, since
the function $F_\gamma$ may not disappear in such points
$(x_{(1,j)},x_{(2,j')},\; (1,j)\in V_1(\gamma),\,
(2,j')\in V_2(\gamma))$ for which there is some pair $(j,j')$ with 
the property $x_{(1,j)}\in A^\e_{s_j}$ and
$x_{(2,j')}\in B^\e_{t_{j'}}$ with some sets $A^\e_{s_j}$ and 
$B^\e_{t_{j'}}$ such that $A^\e_{s_j}=B^\e_{t_{j'}}$, while 
$F_\gamma^\e$ must be zero in such points. On the other hand, in 
the case $|\gamma|=\max(k,l)-\min(k,l)$, i.e. if one of the sets 
$V_1(\gamma)$ or $V_2(\gamma)$ is empty, $F_\gamma=F^\e_\gamma$, \
$Z_\gamma^{(1)}=|\gamma|!Z_{\mu,|\gamma|}(F_\gamma)$, and 
relation~(B7) clearly holds for such diagrams $\gamma$.

In the case $|\gamma|=\max(k,l)-\min(k,l)>0$ such an estimate
will be proved for the probability of the the set where
$F_\gamma\neq F_\gamma^\e$ which implies relation~(B7).

Let us define the sets $A=\bigcupp_{s=1}^{M(\e)} A^\e_s$ and
$B=\bigcupp_{t=1}^{M'(\e)} B^\e_t$. These sets $A$ and $B$ do
not depend on the parameter $\e$. Beside this $\mu(A)<\infty$, and
$\mu(B)<\infty$. Define for all pairs $(j_0,j_0')$ such that
$(1,j_0)\in V_1(\gamma)$, $(2,j_0')\in V_2(\gamma)$ the set
$$
\align
D(j_0,j'_0)&=\{(x_{(1,j)},x_{(2,j')},\; (1,j)\in V_1(\gamma), \,
(2,j')\in V_2(\gamma)) \colon\;\\
&\qquad x_{(1,j_0)}\in A^\e_{s_{j_0}}, \;
x_{(1,j'_0)}\in B^\e_{t_{j'_0}} \quad \text{for some } s_{j_0}
\text{ and } t_{j'_0} \text{ such that }
A^\e_{s_{j_0}}=B^\e_{t_{j'_0}} \\
&\qquad x_{(1,j)}\in A\text{ for all } (1,j)\in V_1(\gamma),
\text{ and }x_{(2,j')}\in B\text{ for all }(2,j')\in V_2(\gamma)\}.
\endalign
$$
Introduce the notation $x^\gamma=(x_{(1,j)},x_{(2,j')}),\,
(1,j)\in V_1(\gamma),\,(2,j')\in V_2(\gamma))$ and put
$D_\gamma
=\{x^\gamma\colon\; F^\e_\gamma(x^\gamma)\neq F_\gamma(x^\gamma)\}$.
The relation
$D_\gamma\subset\bigcupp_{j=1}^k\bigcupp_{j'=1}^l D(j_0,j_0')$
holds, since if
$F^\e_\gamma(x^\gamma)\neq F_\gamma(x^\gamma)$ for some 
vector~$x^\gamma$, then it has some coordinates 
$(1,j_0)\in V_1(\gamma)$ and $(2,j'_0)\in V_2(\gamma)$ such that
$x_{(1,j_0)}\in A^\e_{s_{j_0}}$ and
$x_{(1,j'_0)}\in B^\e_{t_{j'_0}}$ with some sets
$A^\e_{s_{j_0}}=B^\e_{t_{j'_0}}$, and the relation in the last 
line of the definition of $D(j_0,j'_0)$ must also hold for this 
vector $x^\gamma$, since otherwise 
$F_\gamma(x_\gamma)=0=F^\e_\gamma(x_\gamma)$. I claim that there is
some constant $C_1$ such that
$$
\mu^{|V_1(\gamma)|+|V_2(\gamma)|}(D(j_0,j'_0))\le C_1\e
\quad\text{for all sets } D(j_0,j'_0),
$$
where $\mu^{|V_1(\gamma)|+|V_2(\gamma)|}$
denotes the direct product of the measure $\mu$ on some copies of
the original space $(X,\Cal X)$ indexed by $(1,j)\in V_1(\gamma)$
and $(2,j')\in V_2(\gamma)$. To see this relation one has to
observe that $\summ_{A^\e_{s_{j_0}}=B^\e_{t_{j'_0}}}
\mu(A^\e_{s_{j_0}})\mu(B^\e_{t_{j'_0}})\le
\summ\e \mu(A^\e_{s_{j_0}})=\e\mu(A)$. Thus the set $D(j_0,j'_0)$
can be covered by the direct product of a set whose $\mu$ measure is
not greater than $\e\mu(A)$ and of a rectangle whose edges are
either the set $A$ or the set $B$.

The above relations imply that
$$
\mu^{|V_1(\gamma)|+|V_2(\gamma)|}(D_\gamma)\le C_2\e \tag B10
$$
with some constant $C_2>0$.

Relation (B9), estimate (B10), the property c) formulated in
Theorem~10.1 for Wiener--It\^o integrals and the observation that
the function $F_\gamma=F_\gamma(f,g)$ is bounded in supremum norm
if $f$ and $g$ are elementary functions imply the inequality
$$
\align
E\(|\gamma|!Z_{\mu,|\gamma|}(F_\gamma)-
Z^{(1)}_\gamma(\e)\)^2&=
|\gamma!|^2E\( Z_{\mu,|\gamma|}(F_\gamma-F_\gamma^\e)\)^2
\le |\gamma|!\| F_\gamma-F_\gamma^\e\|_2^2 \\
&\le K\mu^{|V_1(\gamma)|+|V_2(\gamma)|}(D_\gamma)\le C\e.
\endalign
$$
This means that relation (B7) holds.

To prove relation (B8) write $E\(Z^{(2)}_\gamma(\e)\)^2$ in the
following form:
$$
\aligned
E\(Z^{(2)}_\gamma(\e)\)^2&={\sum}^\gamma
{\sum}^\gamma c^\e(s_1,\dots,s_k) d^\e(t_1,\dots,t_l)
c^\e(\bar s_1,\dots,\bar s_k) d^\e(\bar t_1, \dots,\bar t_l)\\
&\qquad\qquad  EU(s_1,\dots,s_k,t_1,\dots,t_l,
\bar s_1,\dots,\bar s_k,\bar t_1,\dots,\bar t_l)
\endaligned \tag B11
$$
with
$$ \allowdisplaybreaks
\align
&U(s_1,\dots,s_k,t_1,\dots,t_l,
\bar s_1,\dots,\bar s_k,\bar t_1,\dots,\bar t_l)\\
&\qquad =\prod_{j\colon\; (1,j)\in V_1(\gamma)}\mu_W(A^\e_{s_j})
\prod_{j'\colon\; (2,j')\in V_2(\gamma)} \mu_W(B^\e_{t_{j'}}) \\
&\qquad\qquad \prod_{\bar\jmath\colon\; (1,\bar\jmath)\in V_1(\gamma)}
\mu_W(A^\e_{\bar s_{\bar\jmath}})
\prod_{\bar\jmath'\colon\; (2,\bar\jmath')\in V_2(\gamma)}
\mu_W(B^\e_{\bar t_{\bar\jmath'}}) \\
&\qquad\qquad \biggl[\prod_{j\colon\; (1,j)\in
\{(1,1),\dots,(1,k)\}\setminus V_1(\gamma)}\mu_W(A^\e_{s_j})
\prod_{j'\colon\; (2,j')\in \{(2,1),\dots,(2,l)\}\setminus
\in V_2(\gamma)}\mu_W(B^\e_{t_{j'}}) \\
&\qquad\qquad\qquad -\prod_{j\colon\; (1,j)\in\{(1,1),\dots,(1,k)\}
\setminus V_1(\gamma)}\mu(A^\e_{s_j})\biggr]\\
&\qquad\qquad \biggl[\prod_{\bar\jmath\colon\; (1,\bar\jmath)\in
\{(1,1),\dots,(1,k)\}\setminus V_1(\gamma)}
\mu_W(A^\e_{\bar s_{\bar\jmath}})
\prod_{\bar\jmath'\colon\; (2,\bar\jmath')\in \{(2,1),\dots,(2,l)\}
\setminus\in V_2(\gamma)}\mu_W(B^\e_{\bar t_{\bar \jmath'}}) \\
&\qquad\qquad\qquad -\prod_{\bar\jmath\colon\; (1,j)\in\{(1,1),\dots,(1,k)\}
\setminus V_1(\gamma)}\mu(A^\e_{\bar s_{\bar\jmath}})\biggr]. \tag B12
\endalign
$$
The double sum $\sum^\gamma\sum^\gamma$ in (B11) has to be
understood in the following way. The first summation is taken for
vectors $(s_1,\dots,s_k,t_1,\dots,t_l)$, and these vectors take
such values which were defined in $\sum^\gamma$ in formula (B3).
The second summation is taken for vectors
$(\bar s_1,\dots,\bar s_k,\bar t_1,\dots,\bar t_l)$, and again with
values defined in the summation $\sum^\gamma$.

Relation~(B8) will be proved by means of some estimates about
the expectation of the above defined random variable $U(\cdot)$
which will be presented in the following Lemma~B. Before its
formulation I introduce the following Properties~A and~B.

\medskip\noindent
{\bf Property A.\/} {\it A sequence $s_1,\dots,s_k,t_1,\dots,t_l,
\bar s_1,\dots,\bar s_k,\bar t_1,\dots,\bar t_l$, with elements
$1\le s_j,\bar s_{\bar\jmath}\le M(\e)$, for
$1\le j,\bar\jmath\le k$, and
$1\le t_j,\bar t_{\bar\jmath'}\le M'(\e)$ for
$1\le j',\bar\jmath'\le l$,
satisfies Property~A (depending on a fixed diagram~$\gamma$ and
number~$\e>0$) if the sequences of sets
$\{A^\e_{s_j},B^\e_{t_{j'}},(1,j)\in V_1(\gamma),
(2,j')\in V_2(\gamma)\}$
and
$\{A^\e_{\bar s_{\bar\jmath}},B^\e_{\bar t_{\bar\jmath'}},
(1,\bar\jmath)\in V_1(\gamma),(2,\bar\jmath')\in V_2(\gamma)\}$
agree.
(Here we say that two sequences agree if they contain the same
elements in a possibly different order.)}

\medskip\noindent
{\bf Property B.\/} {\it A sequence $s_1,\dots,s_k,t_1,\dots,t_l,
\bar s_1,\dots,\bar s_k,\bar t_1,\dots,\bar t_l$, with elements
$1\le s_j,\bar s_{\bar\jmath}\le M(\e)$, for
$1\le j,\bar\jmath\le k$, and
$1\le t_j,\bar t_{\bar\jmath'}\le M'(\e)$ for
$1\le j',\bar\jmath'\le l$,
satisfies Property~B (depending on a fixed diagram~$\gamma$ and
number~$\e>0$) if the sequences of sets
$$
\{A^\e_{s_j},B^\e_{t_{j'}},(1,j)\in\{(1,1),\dots,(1,k)\}\setminus
V_1(\gamma),\;(2,j')\in\{(2,1),\dots,(2,l)\}\setminus V_2(\gamma)\}
$$
and
$$
\{A^\e_{\bar s_{\bar\jmath}},B^\e_{\bar t_{\bar\jmath'}},
(1,\bar\jmath)\in\{(1,1),\dots,(1,k)\}\setminus  V_1(\gamma),\;
(2,\bar\jmath')\in\{(2,1),\dots,(2,l)\}\setminus  V_2(\gamma)\}
$$
have at least one common element.}

\medskip
(In the above definitions two sets $A^\e_s$ and $B^\e_t$ are
identified if $A^\e_s=B^\e_t$.)

Now I formulate the following

\medskip\noindent
{\bf Lemma B.} {\it Let us consider the function $U(\cdot)$
introduced in formula (B12). Assume that its arguments
$s_1,\dots,s_k,t_1,\dots,t_l,
\bar s_1,\dots,\bar s_k,\bar t_1,\dots,\bar t_l$ are chosen
in such a way that the function $U(\cdot)$ with these
arguments appears in the double sum $\sum^\gamma\sum^\gamma$
in formula~(B11), i.e.\  $A^\e_{s_j}=B^\e{t_{j'}}$ if 
$((1,j),(2,j'))\in E(\gamma)$, otherwise all sets $A^\e_{s_j}$ and 
$B^\e_{t_{j'}}$ are disjoint, and an analogous statement holds if
the coordinates $s_1,\dots,s_k,t_1,\dots,t_l$ are replaced by
$\bar s_1,\dots,\bar s_k,\bar t_1,\dots,\bar t_l$. Then
$$
EU(s_1,\dots,s_k,t_1,\dots,t_l,
\bar s_1,\dots,\bar s_k,\bar t_1,\dots,\bar t_l)=0 \tag B13
$$
if the sequence of the arguments in $U(\cdot)$ does not satisfies
either Property~A or Property~B.

If the sequence of the arguments in $U(\cdot)$ satisfies both
Property~A and Property~B, then
$$
\aligned
&|EU(s_1,\dots,s_k,t_1,\dots,t_l,
\bar s_1,\dots,\bar s_k,\bar t_1,\dots,\bar t_l)|\\
&\qquad \le C\e \prod{\vphantom\prod}'
\mu(A^\e_{s_j})\mu(A^\e_{\bar s_{\bar\jmath}})
\mu(B^\e_{t_{j'}})\mu(B^\e_{\bar t_{\bar\jmath'}})
\endaligned
\tag B14
$$
with some appropriate constant $C=C(k,l)>0$ depending only on
the number of variables $k$ and $l$ of the functions $f$ and $g$.
The prime in the product $\prod'$ at the right-hand side of~(B14)
means that in this product the measure $\mu$ of those sets
$A^\e_{s_j}$, $A^\e_{\bar s_{\bar\jmath}}$, $B^\e_{t_{j'}}$ and
$B^\e_{\bar t_{\bar\jmath'}}$ are considered, whose indices are
listed among the arguments
$s_j,\bar s_{\bar\jmath},t_{j'}$ or $\bar t_{\bar\jmath'}$ of
$U(\cdot)$, and the measure~$\mu$ of each such set appears exactly 
once. (This means e.g. that if $A^\e_{s_j}=B^\e_{t_{j'}}$ or
$A^\e_{s_j}=B^\e_{\bar t_{\bar\jmath'}}$ for some indices $j$ and 
$j'$ or $\bar\jmath'$, then  one of the terms between 
$\mu(A^\e_{s_j})$ and $\mu(B^\e_{t_{j'}})$ or
$\mu(B^\e_{\bar t_{\bar\jmath'}})$ is omitted from the product. 
For the sake of definitiveness let us preserve the set 
$\mu(A^\e_{j_s})$ in such a case.)}

\medskip\noindent
{\it Remark.}\/ The content of Lemma~B is that most terms in the
double sum in formula~(B11) equal zero, and even the non-zero 
terms are small.

\medskip\noindent
{\it The proof of Lemma B.}\/ Let us prove first relation (B13)
in the case when Property~A does not hold. It will be exploited 
that for disjoint sets the random variables $\mu_W(A_s)$ and 
$\mu_W(B_t)$ are independent, and this provides a good 
factorization of the expectation of certain products. Let us 
carry out the multiplications in the definition of $U(\cdot)$ 
in formula~(B12), and show that each product obtained in such a 
way has zero expectation. If Property~A does not hold for the 
arguments of $U(\cdot)$, and beside this the arguments 
$s_1,\dots,s_k,t_1,\dots,t_l,
\bar s_1,\dots,\bar s_k,\bar t_1,\dots,\bar t_l$ satisfy the
remaining conditions of Lemma~B, then each
product we consider contains a factor $\mu_W(A^\e_{s_{j_0}})$,
$(1,j_0)\in V_1(\gamma)$, which is independent of all those terms
in this product which are in the following list: $\mu_W(A^\e_{s_j})$
with some $j\neq j_0$, $1\le j\le k$, or $\mu_W(B^\e_{t_{j'}})$, 
$1\le j\le l$, or $\mu_W(A^\e_{\bar s_{\bar\jmath}})$ with
$(1,\bar\jmath)\in V_1(\gamma)$, or 
$\mu_W(B^\e_{\bar t_{\bar\jmath'}})$ with 
$(2,\bar\jmath')\in V_2(\gamma)$. We will show with the help of 
this property that the expectation of each term has a factorization 
with a factor either of the form $E\mu_W(A^\e_{s_{j_0}})=0$ or 
$E\mu_W(A^\e_{s_{j_0}})^3=0$, hence it equals zero. 
Indeed, although the above  properties do not exclude the 
appearance of such a pair of  arguments $A^\e_{t_{\bar j'}}$, 
$(1,\bar\jmath')\in\{(1,1),\dots,(1,k)\setminus V_1(\gamma)$ and 
$B^\e_{t_{\bar j'}}$, $(2,\bar\jmath')\in\{(2,1),\dots,(2,l)\}
\setminus V_2(\gamma)$ in the product for which
$A^\e_{t_{\bar j}}=B^\e_{t_{\bar j'}}=A^\e_{s_{j_0}}$, and in 
such a case a term of the form $E\mu_W(A^\e_{s_{j_0}})$ will not 
appear in the product, but if this happens, then the product contains 
a factor of the form $E\mu_W(A^\e_{s_{j_0}})^3=0$. Hence an 
appropriate factorization of each term of $EU(\cdot)$
contains either a factor of the form
$E\mu_W(A^\e_{s_{j_0}})=0$ or $E\mu_W(A^\e_{s_{j_0}})^3=0$ if 
$U(\cdot)$ does not satisfy Property~A.  

To finish the proof of relation (B13) it is enough consider the
case when the arguments of $U(\cdot)$ satisfy Property~A, but they
do not satisfy Property~B. The validity of Property~A implies that
the sets $\{A^\e_{s_j},\,j\in V_1\}\cup\{B^\e_{t_{j'}},\,j'\in V_2\}$ 
and 
$\{A^\e_{\bar s_j},\,j\in V_1\}\cup\{B^\e_{\bar t_{j'}},\,j'\in V_2\}$ 
agree. The conditions of Lemma~B also imply that the elements
of these sets are such sets which are disjoint of the sets
$A^\e_{s_j}$, $B^\e_{t_{j'}}$, $A^\e_{\bar s_{\bar\jmath}}$ and 
$B^\e_{\bar t_{\bar\jmath'}}$ with indices 
$(1,j),(1,\bar\jmath)\in\{(1,1),\dots,(1,k)\}\setminus V_1(\gamma)$ 
and 
$(2,j'),(2,\bar\jmath')\in\{(2,1),\dots,(2,l)\}\setminus V_2(\gamma)$.
If Property~B does not hold, then the latter class of sets can be 
divided into two subclasses in such a way that the elements in 
different subclasses are disjoint. The first subclass consists of the 
sets $A^\e_{s_j}$ and $B^\e_{t_{j'}}$, and the second one of the sets 
$A^\e_{\bar s_{\bar\jmath}}$ and $B^\e_{\bar t_{\bar\jmath'}}$
with indices such that
$(1,j),(1,\bar\jmath)\in\{(1,1),\dots,(1,k)\}\setminus V_1(\gamma)$ and
$(2,j'),(2,\bar\jmath')\in\{(2,1),\dots,(2,l)\}\setminus V_2(\gamma)$.
These facts imply that $EU(\cdot)$ has a factorization, which contains 
the term
$$ \allowdisplaybreaks
\align
&E\biggl[\prod_{j\colon\; (1,j)\in
\{(1,1),\dots,(1,k)\}\setminus V_1(\gamma)}\mu_W(A^\e_{s_j})
\prod_{j'\colon\; (2,j')\in \{(2,1),\dots,(2,l)\}\setminus
\in V_2(\gamma)}\mu_W(B^\e_{t_{j'}}) \\
&\qquad\qquad -\prod_{j\colon\; (1,j)\in\{(1,1),\dots,(1,k)\}
\setminus V_1(\gamma)}\mu(A^\e_{s_j})\biggr]=0,
\endalign
$$
hence relation (B13) holds also in this case. The last expression
has zero expectation, since if we take such pairs 
$A^\e_{s_j},B^\e_{t_j'}$ for the sets appearing in it for which
that $((1,j),(2,j'))\in E(\gamma)$, i.e. these vertices are
connected with an edge of $\gamma$, then $A^\e_{s_j}=B^\e_{t_j'}$ 
in a pair, and elements in different pairs are disjoint. This 
observation allows a factorization in the product whose 
expectation is taken, and then the identity
$E\mu_W(A^\e_{s_j})\mu_W(B^\e_{t_{j'}})=\mu(A^\e_{s_j})$ implies
the desired identity.

To prove relation (B14) if the arguments of the function~$U(\cdot)$
satisfy both Properties~A and~B consider the expression (B12) which
defines $U(\cdot)$, carry out the term by term multiplication
between the two differences at the end of this formula, take
expectation for each term of the sum obtained in such a way, and
factorize them. Since $E\mu_W(A)^2=\mu(A)$,
$E\mu_W(A)^4=3\mu(A)^2$ for all sets $A\in\Cal X$, $\mu(A)<\infty$,
some calculation shows that each term can be expressed as  constant
times a product whose elements are those probabilities $\mu(A_s^\e)$
and $\mu(B_t^\e)$ or their square which appear at the right-hand
side of (B14). Moreover, since the arguments of $U(\cdot)$ satisfy
Property~B, there will be at least one term of the form
$\mu(A_s^\e)^2$ in this product. Since
$\mu(A_s^\e)^2\le \e\mu(A_s^\e)$, these calculations provide
formula~(B14). Lemma~B is proved.

\medskip
Relation (B11) implies that
$$
E\(Z^{(2)}_\gamma(\e)\)^2\le K\sum{\vphantom{\sum}}^\gamma
\sum{\vphantom{\sum}}^\gamma
|EU(s_1,\dots,s_k,t_1,\dots,t_l,
\bar s_1,\dots,\bar s_k,\bar t_1,\dots,\bar t_l)| \tag B15
$$
with some appropriate $K>0$. By Lemma~B it is enough to sum up
only for such terms $U(\cdot)$ in (B15) whose arguments satisfy
both Properties~A and~B. Moreover, each such term can be bounded
by means of inequality (B14). Let us list the sets
$A^\e_{s_j},A^\e_{\bar s_{\bar\jmath}}, B^\e_{t_{j'}},
B^\e_{\bar t_{\bar\jmath'}}$ appearing in the
upper bound at the right-hand side of (B14) for
all functions $U(\cdot)$ taking part in the sum at the right-hand
side of (B15). Since all fixed sequences of the sets $A^\e_s$ and
$B^\e_t$ appear less than $C(k,l)$ times with an appropriate
constant $C(k,l)$ depending only on the order $k$ and $l$ of the
integrals we are considering, and
$\summ_{s=1}^{M(\e)}\mu(A^\e_s)+\summ_{t=1}^{M'(\e)}\mu(B^\e_t)
=\mu(A)+\mu(B)<\infty$, the above relations imply that
$$
E\(Z^{(2)}_\gamma(\e)\)^2\le C_1\e\sum_{j=1}^{k+l}(\mu(A)+\mu(B))^j
\le C\e.
$$
Hence relation (B8) holds.

\medskip
To prove Theorem 10.2A in the general case take for all pairs of 
functions $f\in\Cal H_{\mu,k}$ and $g\in\Cal H_{\mu,l}$ two 
sequences of elementary functions $f_n\in\bar{\Cal H}_{\mu,k}$ 
and $g_n\in\bar{\Cal H}_{\mu,l}$, $n=1,2,\dots$, such that
$\|f_n-f\|_2\to0$ and $\|g_n-g\|_2\to0$ as $n\to\infty$. Let us
introduce the notation $F_\gamma(f,g)=F_\gamma$ if the function
$F_\gamma$ is defined in formulas~(10.9a) and~(10.10) with the 
help of the functions $f$ and~$g$. It is enough to show that
$$
E|k!Z_{\mu,k}(f)l!Z_{\mu,l}(g)-k!Z_{\mu,k}(f_n)
l!Z_{\mu,l}(g_n)|\to0\quad \text{as }n\to\infty,
\tag B16
$$
and
$$
|\gamma|!E\left|Z_{\mu,|\gamma|}(F_\gamma(f,g))-
Z_{\mu,|\gamma|}(F_\gamma(f_n,g_n))\right|\to0\quad \text{as }
n\to\infty \quad\text{for all } \gamma\in\Gamma(k,l), \tag B17
$$
since then a simple limiting procedure $n\to\infty$, and the
already proved part of the theorem for Wiener--It\^o integrals of
elementary functions imply Theorem~10.2A.

To prove relation (B16) write
$$
\align
&E|k!Z,{\mu,k}(f)l!Z_{\mu,l}(g)-
k!Z_{\mu,k}(f_n)l!Z_{\mu,l}(g_n)|\\
&\qquad\le k!l!\(E|Z_{\mu,k}(f)Z_{\mu,l}(g-g_n)|
+E|Z_{\mu,k}(f-f_n)Z_{\mu,l}(g_n)\)| \\
&\qquad\le k!l!
\(\(EZ^2_{\mu,k}(f)\)^{1/2}\(EZ^2_{\mu,l}(g-g_n)\)^{1/2}+
\(EZ^2_{\mu,k}(f-f_n)\)^{1/2}\(EZ^2_{\mu,l}(g_n)\)^{1/2}\)\\
&\qquad\le (k!l!)^{1/2}\(\|f\|_2\|g-g_n\|_2+\|f-f_n\|_2\|g_n\|_2\).
\endalign
$$
Relation (B16) follows from this inequality with a limiting procedure
$n\to\infty$.

To prove relation (B17) write
$$
\align
&|\gamma|!E\left|Z_{\mu,|\gamma|}(F_\gamma(f,g))-
Z_{\mu,|\gamma|}(F_\gamma(f_n,g_n))\right|\\
&\qquad\le
|\gamma|!E\left|Z_{\mu,|\gamma|}(F_\gamma(f,g-g_n))\right|+
|\gamma|!E\left|Z_{\mu,|\gamma|}(F_\gamma(f-f_n,g_n))\right|\\
&\qquad\le
|\gamma|!\left(EZ^2_{\mu,|\gamma|}(F_\gamma(f,g-g_n))\right)^{1/2}+
|\gamma|!\left(EZ^2_{\mu,|\gamma|}(F_\gamma(f-f_n,g_n))\right)^{1/2}\\
&\qquad\le (|\gamma|!)^{1/2}\(\|F_\gamma(f,g-g_n)\|_2+
\|F_\gamma(f-f_n,g_n)\|_2\),
\endalign
$$
and observe that by relation (10.11)
$\|F_\gamma(f,g-g_n)\|_2\le \|f\|_2\|g-g_n\|_2$, and
$\|F_\gamma(f-f_n,g_n)\|_2\le \|f-f_n\|_2\|g_n\|_2$. Hence
$$
\align
&|\gamma|!E\left|Z_{\mu,|\gamma|}(F_\gamma(f,g))-
Z_{\mu,|\gamma|}(F_\gamma(f_n,g_n))\right|\\
&\qquad\le(|\gamma|!)^{1/2}
\(\|f\|_2\|g-g_n\|_2+\|f-f_n\|_2\|g_n\|_2\).
\endalign
$$
The last inequality implies relation (B17) with a limiting procedure
$n\to\infty$. Theorem 10.2A is proved.

\beginsection Appendix C. The proof of some results about
Wiener--It\^o integrals.

First I prove It\^o's formula about multiple Wiener--It\^o integrals
(Theorem~10.3). The proof is based on the diagram formula for
Wiener--It\^o integrals and a recursive formula about Hermite 
polynomials proved in Proposition~C. In Proposition~C2 I present the
proof of another important property of Hermite polynomials. This 
result states that the class of all Hermite polynomials is a 
{\it complete} orthogonal system in an appropriate Hilbert space. It
is needed in the proof of Theorem 10.5 about the isomorphism of Fock 
spaces to the Hilbert space generated by Wiener--It\^o 
integrals. At the end of Appendix~C the proof of Theorem~10.4, a
limit theorem about degenerated $U$-statistics is given. 

\medskip\noindent
{\bf Proposition C about some properties of Hermite polynomials.}
{\it The functions
$$
H_k(x)=(-1)^k e^{x^2/2}\frac {d^k}{dx^k}e^{-x^2/2},
\quad k=0,1,2,\dots \tag C1
$$
are the Hermite polynomials with leading coefficient 1, i.e.\ $H_k(x)$
is a polynomial of order $k$ with leading coefficient 1 such that
$$
\int_{-\infty}^\infty H_k(x)H_l(x) \frac1{\sqrt{2\pi}}e^{-x^2/2}\,dx=0
\quad \text{if } k\neq l, \tag C2
$$
and
$$
\int_{-\infty}^\infty H^2_k(x) \frac1{\sqrt{2\pi}}e^{-x^2/2}\,dx=k!
\quad \text{for all } k=0,1,2\dots. \tag C$2'$
$$
The recursive relation
$$
H_k(x)=x H_{k-1}(x)-(k-1)H_{k-2}(x) \tag C3
$$
holds for all $k=1,2,\dots$.}

\medskip\noindent
{\it Remark.} It is more convenient to consider relation (C3) valid
also in the case $k=1$. In this case $H_1(x)=x$, $H_0(x)=1$, and
relation holds with an arbitrary function $H_{-1}(x)$.

\medskip\noindent
{\it Proof of Proposition C.} It is clear from formula (C1) that
$H_k(x)$ is a polynomial of order $k$ with leading coefficient 1.
Take $l\ge k$, and write by means of integration by parts
$$
\align
\int_{-\infty}^\infty H_k(x)H_l(x) \frac1{\sqrt{2\pi}}e^{-x^2/2}\,dx&=
\int_{-\infty}^\infty\frac1{\sqrt{2\pi}} H_k(x)(-1)^l\frac{d^l}{dx^l}
e^{-x^2/2}\,dx\\
&=\int_{-\infty}^\infty\frac1{\sqrt{2\pi}} \frac d{dx} H_k(x)
(-1)^{l-1}\frac{d^{l-1}}{dx^{l-1}}e^{-x^2/2}\,dx.
\endalign
$$
Successive partial integration together with the identity
$\frac{d^k}{dx^k}H_k(x)=k!$ yield that
$$
\int_{-\infty}^\infty H_k(x)H_l(x) \frac1{\sqrt{2\pi}}e^{-x^2/2}\,dx=
k!\int_{-\infty}^\infty\frac1{\sqrt{2\pi}}
(-1)^{l-k}\frac{d^{l-k}}{dx^{l-k}}e^{-x^2/2}\,dx.
$$
The last relation supplies formulas (C2) and (C$2'$).

To prove relation (C3) observe that $H_k(x)-xH_{k-1}(x)$ is a
polynomial of order $k-2$. (The term $x^{k-1}$ is missing from this
expression. Indeed, if $k$ is an even number, then the polynomial
$H_k(x)-xH_{k-1}(x)$ is an even function, and it does not contain
the term $x^{k-1}$ with an odd exponent $k-1$. Similar argument
holds if the number $k$ is odd.) Beside this, it is orthogonal
(with respect to the standard normal distribution) to all Hermite
polynomials $H_l(x)$ with $0\le l\le k-3$. Hence
$H_k(x)-xH_{k-1}(x)=CH_{k-2}(x)$ with some constant $C$ to be
determined.

Multiply both sides of the last identity with $H_{k-2}(x)$
and integrate them with respect to the standard normal
distribution. Apply the orthogonality of the polynomials
$H_k(x)$ and $H_{k-2}(x)$, and observe that the identity
$$
\int H_{k-1}(x)xH_{k-2}(x)\frac1{\sqrt{2\pi}}e^{-x^2/2}\,dx=
\int H^2_{k-1}(x)\frac1{\sqrt{2\pi}}e^{-x^2/2}\,dx=(k-1)!
$$
holds. (In this calculation we have exploited that $H_{k-1}(x)$
is orthogonal to $H_{k-1}(x)-xH_{k-2}(x)$, because the order of
the latter polynomial is less than $k-1$.) In such a way we get
the identity $-(k-1)!=C(k-2)!$ for the constant~$C$ in the last
identity, i.e. $C=-(k-1)$, and this implies relation (C3).

\medskip\noindent
{\it Proof of It\^o's formula for multiple Wiener--It\^o
integrals.}\/ Let $K=\summ_{p=1}^m k_p$, the sum of the order of
the Hermite polynomials, denote the order of the expression in
relation (10.20). Formula (10.20) clearly holds for expressions of
order $K=1$. It will be proved in the general case by means of
induction with respect to the order~$K$.

In the proof the functions $f(x_1)=\varphi_1(x_1)$ and
$$
g(x_1,\dots,x_{K_m-1})=\prod_{j=1}^{K_1-1}\varphi_1(x_j)
\cdot \prod_{p=2}^m \prod_{j=K_{p-1}}^{K_p-1}\varphi_p(x_j),
$$
will be introduced and the product
$Z_{\mu,1}(f)(K_m-1)!Z_{\mu,K_m-1}(g)$ will be calculated by means
of the diagram formula. (The same notation is applied as in
Theorem 10.3. In particular, $K=K_m$, and in the case $K_1=1$ the
convention $\prodd_{j=1}^{K_1-1}\varphi_1(x_j)=1$ is applied.)
In the application of the diagram formula diagrams with two rows
appear. The first row of these diagrams contains the
vertex $(1,1)$ and the second row contains the vertices
$(2,1),\dots,(2,K_m-1)$. It is useful to divide the diagrams to
three disjoint classes. The first class contains only the diagram
$\gamma_0$ without any edges. The second class $\Gamma_1$
consists of those diagrams which have an edge of the form
$((1,1),(2,j))$ with some $1\le j\le k_1-1$, and the third class
$\Gamma_2$ is the set of those diagrams which have an edge of the
form $((1,1),(2,j))$ with some $k_1\le j\le K_m-1$.
Because of the orthogonality of the functions $\varphi_s$ for
different indices~$s$ $F_\gamma\equiv0$ and
$Z_{\mu,K_m-2}(F_\gamma)=0$ for $\gamma\in\Gamma_2$.
The class $\Gamma_1$ contains $k_1-1$ diagrams. Let us consider a
diagram $\gamma$ from this class with an edge $((1,1),(2,j_0))$,
$1\le j\le k_1-1$. We have for such a diagram $F_\gamma=
\prodd_{j\in\{1,\dots,K_1-1\}\setminus \{j_0\}}\varphi_1(x_{(2,j)})
\prodd_{p=2}^m \prodd_{j=K_{p-1}}^{K_p-1}\varphi_p(x_{(2,j)})$, and
by our inductive hypothesis $(K_m-2)!Z_{\mu,K_m-2}(F_\gamma)=
H_{k_1-2}(\eta_1)\prodd_{p=2}^m H_{k_p}(\eta_p)$. Finally
$$
K_m!Z_{\mu,K_m}(F_{\gamma_0})=
K_m!Z_{\mu,K_m}\(\prod_{p=1}^m \(\prod_{j=K_{p-1}+1}^{K_p}
\varphi_p(x_j)\)\)
$$
for the diagram $\gamma_0$ without any edge.

Our inductive hypothesis also implies the following identity for
the expression we wanted to calculate with the help of the diagram
formula.
$$
Z_{\mu,1}(f)(K_m-1)!Z_{\mu,K_m-1}(g)=\eta_1
H_{k_1-1}(\eta_1)\prodd_{p=2}^m H_{k_p}(\eta_p).
$$

The above calculations together with the observation 
$|\Gamma_1|=k_1-1$ yield the identity
$$ \allowdisplaybreaks
\align
&K_m!Z_{\mu,K_m}\(\prod_{p=1}^m \(\prod_{j=K_{p-1}+1}^{K_p}
\varphi_p(x_j)\)\)=K_m!Z_{\mu,K_m}(F_{\gamma_0})\\
&\qquad=Z_{\mu,1}(f)(K_m-1)!Z_{\mu,K_m-1}(g)-
\sum_{\gamma\in\Gamma_1}(K_m-2)!Z_{\mu,K_m-2}(F_\gamma)\\
&\qquad=\eta_1 H_{k_1-1}(\eta_1)\prod_{p=2}^m H_{k_p}(\eta_p)
-(k_1-1) H_{k_1-2}(\eta_1)\prod_{p=2}^m H_{k_p}(\eta_p)\\
&\qquad=\[\eta_1H_{k_1-1}(\eta_1)
-(k_1-1) H_{k_1-2}(\eta_1)\]\prod_{p=2}^m H_{k_p}(\eta_p).
\tag C4
\endalign 
$$
On the other hand, $\eta_1 H_{k_1-1}(\eta_1)
-(k_1-1) H_{k_1-2}(\eta_1)=H_{k_1}(\eta_1)$ by formula (C3).
These relations imply formula (10.20), i.e. It\^o's formula.

\medskip
I present the proof of another important property of the Hermite
polynomials in the following Proposition~C2.

\medskip\noindent
{\bf Proposition~C2 on the completeness of the orthogonal system
of Hermite polynomials.} {\it The Hermite polynomials $H_k(x)$,
$k=0,1,2,\dots$, defined in formula (C4) constitute a complete
orthonormal system of in the $L_2$-space of the functions square
integrable with respect to the Gaussian measure
$\frac1{\sqrt{2\pi}}e^{-x^2/2}\,dx$ on the real line.}

\medskip\noindent
{\it Proof of Proposition C2.} Let us consider the orthogonal
complement of the subspace generated by the Hermite polynomials
in the space of the square integrable functions with respect
to the measure $\frac1{\sqrt{2\pi}}e^{-x^2/2}\,dx$. It is enough
to prove that this orthogonal completion contains only the
identically zero function. Since the orthogonality of a function to
all polynomials of the form $x^k$, $k=0,1,2,\dots$ is equivalent
to the orthogonality of this function to all Hermite polynomials
$H_k(x)$, $k=0,1,2,\dots$, Proposition~C2 can be reformulated in
the following form:

If a function $g(x)$ on the real line is such that
$$
\int_{-\infty}^\infty x^k g(x)\frac1{\sqrt{2\pi}}e^{-x^2/2}\,dx=0
\quad \text{for all }k=0,1,2,\dots \tag C5
$$
and
$$
\int_{-\infty}^\infty g^2(x)\frac1{\sqrt{2\pi}}e^{-x^2/2}\,dx<\infty,
\tag C6
$$
then $g(x)=0$ for almost all $x$.

Given a function $g(x)$ on the real line whose absolute value is
integrable with respect to the Gaussian measure 
$\frac1{\sqrt{2\pi}}e^{-x^2/2}\,dx$ define the (finite) 
measure $\nu_g$,
$$
\nu_g(A)=\int_A g(x)\frac1{\sqrt{2\pi}}e^{-x^2/2}\,dx
$$
on the measurable sets of the real line together with  its Fourier 
transform $\tilde\nu_g(t)=\int_{-\infty}^\infty e^{itx}\nu_g(\,dx)$. 
(This measure $\nu_g$ and its Fourier transform can
be defined for all functions~$g$ satisfying relation (C6), because
their absolute value  is integrable with respect to the Gaussian 
measure.) First I show that Proposition~C2 can be reduced to the 
following statement: If a function $g$ satisfies both (C5) and (C6) 
then $\tilde\nu_g(t)=0$ for all $-\infty<t<\infty$.

Indeed, if there were a function $g$ satisfying (C5) and (C6) which
is not identically zero, then the non-negative functions
$g^+(x)=\max(0,g(x))$ and $g^-(x)=-\min(0,g(x))$ would be different.
Then also their Fourier transform $\tilde\nu_{g^+}(t)$ and
$\tilde\nu_{g^-}(t)$ would be different, since a finite measure is
uniquely determined by its Fourier transform. (This statement is
equivalent to an important result in probability  theory, by which
a probability measure on the real line is determined by
its characteristic function.) But this would mean that
$\tilde\nu_{g}(t)=\tilde\nu_{g^+}(t)-\tilde\nu_{g^-}(t)\neq0$ for
some~$t$. Hence Proposition~C2 can be reduced to the above
statement.

Since $\left|e^{itx}-1-(itx)-\cdots-\frac{(itx)^k}{k!}\right|\le
\frac{|tx|^{(k+1)}}{(k+1)!}$ for all real numbers $t$, $x$ and
integer $k=1,2,\dots$ we may write because of relation~(C5)
$$
\align
|\tilde\nu_g(t)|&=\left|\int_{-\infty}^\infty
\(e^{itx}-1-(itx)-\cdots-\frac{(itx)^k}{k!}\)g(x)
\frac1{\sqrt{2\pi}}e^{-x^2/2}\,dx\right| \\
&\le\int_{-\infty}^\infty \frac{|t|^{(k+1)}}{(k+1)!}|x|^{k+1}|g(x)|
\frac1{\sqrt{2\pi}}e^{-x^2/2}\,dx
\endalign
$$
for all $k=1,2,\dots$ and real number $t$ if the function $g$
satisfies relation~(C5). If it satisfies both relation (C5)
and~(C6), then from the last relation and the Schwarz inequality
$$
\align
|\tilde\nu_g(t)|^2&\le\const\frac{|t|^{2(k+1)}}{\((k+1)!\)^2}
\int_{-\infty}^\infty |x|^{2(k+1)}\frac1{\sqrt{2\pi}}e^{-x^2/2}\,dx\\
&=\const \frac{|t|^{2(k+1)}}{\((k+1)!\)^2}1\cdot3\cdot5\cdots(2k+1)
\endalign
$$
for all real number $t$ and integer $k=1,2,\dots$. Simple
calculation shows that the right-hand side of the last estimate
tends to zero as $k\to\infty$. This implies that $\tilde\nu_g(t)=0$
for all $t$, and Proposition~C2 holds.

\medskip
I finish Appendix~C with the proof of Theorem 10.4, a limit
theorem about a sequence of normalized degenerate $U$-statistics.
It is based on an appropriate representation of the
$U$-statistics by means of multiple random integrals which makes
possible to carry out an appropriate limiting procedure.

\medskip\noindent
{\it Proof of Theorem 10.4.}
For all $n=1,2,\dots$, the normalized degenerate $U$-statistics
$n^{-k/2}I_{n,k}(f)$ can be written in the form
$$
\aligned
n^{-k/2}k!I_{n,k}(f)&=n^{k/2}\int'
f(x_1,\dots,x_k)\mu_n(\,dx_1)\dots\mu_n(\,dx_k)\\
&=n^{k/2}\int' f(x_1,\dots,x_k)(\mu_n(\,dx_1)-\mu(\,dx_1))
\dots(\mu_n(\,dx_k)-\mu(\,dx_1)),
\endaligned \tag C7
$$
where $\mu_n$ is the empirical distribution function of the
sequence $\xi_1,\dots,\xi_n$ defined in~(4.5), and the prime
in $\int'$ denotes that the diagonals, i.e. the points
$x=(x_1,\dots,x_k)$ such that $x_j=x_{j'}$ for some pairs
of indices $1\le j,j'\le k$, $j\neq j'$, are omitted from the
domain of integration. The second identity in relation (C7) can
be justified by means of the identity
$$
\aligned
& \int'f(x_1,\dots,x_k)
(\mu_n(\,dx_1)-\mu(\,dx_1))\dots(\mu_n(\,dx_k)-\mu(\,dx_1))-
I_{n,k}(f) \\
&\qquad =\sum _{V\colon\; V\in\{1,\dots,k\},\,|V|\ge 1}(-1)^{|V|}
\int'f(x_1,\dots,x_k)\prod_{j\in V}\mu(\,dx_j)
\prod_{j\in\{1,\dots, k\}\setminus V}\mu_n(\,dx_j))=0.
\endaligned \tag C8
$$
This identity holds for a function $f$ canonical with respect
to a non-atomic measure $\mu$, because each term in  the sum at
the right-hand side of (C8) equals zero. Indeed, the integral of
a canonical function $f$ with respect to $\mu(\,dx_j)$ with some
index $j\in V$ equals zero for all fixed values
$x_1,\dots,x_{j-1},x_{j+1},\dots,x_k$. The non-atomic
property of the measure $\mu$ was needed to guarantee that this
integral equals zero also in the case when the diagonals are
omitted from the domain of integration.

We would like to derive Theorem 10.4 from relation (C7) by
means of an appropriate limiting procedure which exploits the
convergence of the random fields $n^{1/2}(\mu_n(A)-\mu(A))$,
$A\in \Cal X$, to a Gaussian field $\nu(A)$, $A\in\Cal X$, as
$n\to\infty$. But some problems arise if we want to carry out
such a program, because the fields $n^{1/2}(\mu_n-\mu)$ converge
to a non white noise type Gaussian field. The limit we get is
similar to a Wiener bridge on the real line. Hence a relation
between Wiener processes and Wiener bridges suggests to write
the following version of formula (C7).

Let us take a standard Gaussian random variable $\eta$, independent
of the random sequence $\xi_1,\xi_2,\dots$. For a canonical
function~$f$ the following version of (C7) holds.
$$
n^{-k/2}k!I_{n,k}(f)=J'_{n,k}(f) \tag C9
$$
with
$$
\aligned
J'_{n,k}(f)=\int'
&f(x_1,\dots,x_k)\[\sqrt n(\mu_n(\,dx_1)-\mu(\,dx_1))
+\eta\mu(\,dx_1)\]\\
&\qquad\dots\[\sqrt n(\mu_n(\,dx_k)-\mu(\,dx_k))+\eta\mu(\,dx_k)\].
\endaligned \tag C10
$$
This relation can be seen similarly to (C7).

The random measures $n^{1/2}(\mu_n-\mu)+\eta\mu$ converge to
a white noise with reference measure $\mu$. Hence Theorem~10.4 can
be proved by means of formulas~(C9) and~(C10) with the help of an
appropriate limiting procedure. More explicitly, I claim that
the following slightly more general result holds. The expressions
$J'_{n,k}(f)$ introduced in (C10) converge in distribution to
the Wiener--It\^o integral $k!Z_{\mu,k}(f)$ as $n\to\infty$ for all
functions $f$ square integrable with respect to the product measure
$\mu^k$. This result also holds for non-canonical functions~$f$. 
This limit theorem together with relation~(C9) imply Theorem 10.4.

The convergence of the random variables $J'_{n,k}(f)$
defined in (C10) to the Wiener--It\^o integral $k!Z_{\mu,k}(f)$
can be easily checked for elementary functions $f\in \bar
{\Cal H}_{\mu,k}$. Indeed, if $A_1,\dots, A_M$ are disjoint sets
with $\mu(A_s)<\infty$, then the multi-dimensional central limit
theorem implies that the random vectors $\{\sqrt n
((\mu_n(A_s)-\mu(A_s))+\eta\mu(A_s),\,1\le s \le M\}$ converge
in distribution to the random vector
$\{(\mu_W(A_s),\,1\le s\le M\}$, i.e. to a set of independent
normal random variables $\zeta_s$, $E\zeta_s=0$, $1\le s\le M$,
with variance $E\zeta_s^2=\mu(A_s)$ as $n\to\infty$. The
definition of the elementary functions given in (10.2) shows that
this central limit theorem implies the demanded convergence of the
sequence $J'_{n,k}(f)$ to $k!Z_{\mu,k}(f)$ for elementary functions.

To show the convergence of the sequence $J'_{n,k}(f)$ to
$k!Z_{\mu,k}(f)$ in the general case, take for any function
$f\in\Cal H_{\mu,k}$ a sequence of elementary functions
$f_N\in{\bar\Cal H}_{\mu,k}$ such that $\|f-f_N\|_2\to0$
as $N\to\infty$. Then $E(Z_{\mu,k}(f)-Z_{\mu,k}(f_N))^2
=E(Z_{\mu,k}(f-f_N))^2\to 0$ as $N\to\infty$ by Property~c)
in Theorem~10.1. Hence the already proved part of the theorem
implies that there exists some sequence of positive integers,
$N(n)$, $n=1,2,\dots$, in such a way that $N(n)\to\infty$, and
the sequence $J'_{n,k}(f_{N(n)})$ converges to $k!Z_{\mu,k}(f)$
in distribution as $n\to\infty$. Thus to complete the proof of 
Theorem~10.4 it is enough to show that 
$E(J'_{n,k}(f_{N(n)})-J'_{n,k}(f))^2=
E(J'_{n,k}(f_{N(n)}-f))^2\to0$ as $n\to\infty$.

It is enough to show that
$$
E(J'_{n,k}(f))^2\le C\|f\|_2^2\quad\text{for all }f\in\Cal H_{\mu,k}
\tag C11
$$
with a constant $C=C_k$ depending only on the order $k$ of the
function $f$ and to apply inequality~(C11) for the functions
$f_{N(n)}-f$. Relation~(C11) is a relatively simple consequence of
Corollary~1 of Theorem~9.4.

Indeed,
$$
J'_{n,k}(f)=\sum_{V\subset\{1,\dots,k\}}
\eta^{k-|V|} |V|! J_{n,|V|}(f_V)
$$
with
$$
f_V(x_j,\,j\in V)=\int f(x_1,\dots,x_k)\prod_{j'\in
\{1,\dots,k\}\setminus V}\,\mu(dx_{j'})
$$
and the random integral $J_{n,k}(\cdot)$ defined in (4.8), hence
$$
E(J'_{n,k}(f))^2\le 2^k \sum_{V\subset\{1,\dots,k\}}
(|V|!)^2 E\eta^{2(k-|V|)}\cdot EJ^2_{n,|V|}(f_V). \tag C12
$$
Inequality $\|f_V\|_2\le \|f\|_2$ holds for all sets
$V\subset\{1,\dots,k\}$, hence an application of Corollary~1 of
Theorem~(9.4) to all random integrals $J_{n,|V|}(f)$ supplies~(C11).

The above proof also yields the following slight generalization of
Theorem~10.4. Let us consider a finite sequence of functions
$f_j\in \Cal H_{\mu,j}$, $1\le j\le k$, canonical with respect to
a non-atomic probability measure $\mu$. The vectors
$\{n^{-j/2}I_{n,j}(f_j),1\le j\le k\}$, consisting of normalized
degenerate $U$-statistics defined with the help of a sequence of
independent $\mu$-distributed random variables converge to the
random vector $\{Z_{\mu,j}(f_j),1\le j\le k\}$ in distribution as
$n\to\infty$. This result together with Theorem~9.4 imply the
following limit theorem about multiple random
integrals~$J_{n,k}(f)$.

\medskip\noindent
{\bf Theorem 10.4$'$. (Limit theorem about multiple random integrals
with respect to normalized empirical measures).} {\it Let a sequence
of independent and identically distributed random variables
$\xi_1,\xi_2,\dots$ be given with some non-atomic distribution
$\mu$ on a measurable space $(X,\Cal X)$ together with a function
$f(x_1,\dots,x_k)$ on the $k$-fold product $(X^k,\Cal X^k)$ of the
space $(X,\Cal X)$ such that
$$
\int f^2(x_1,\dots,x_k)\mu(\,dx_1)\dots\mu(\,dx_k)<\infty.
$$
Let us consider for all $n=1,2,\dots$ the random integrals
$J_{n,k}(f)$ of order~$k$ defined in formulas (4.5) and (4.8) with
the help of the empirical distribution $\mu_n$ of the sequence
$\xi_1,\dots,\xi_n$ and the function~$f$. These random integrals
$J_{n,k}(f)$ converge in distribution, as $n\to\infty$, to the
following sum $U(f)$ of multiple Wiener--It\^o integrals:
$$
\align
U(f)&= \sum_{V\subset\{1,\dots,k\}} C(k,V)Z_{\mu,|V|}(f_V)\\
&=\sum_{V\subset\{1,\dots,k\}} \frac{C(k,V)}{|V|!}
\int f_V(x_j,\,j\in V)\prod_{j\in V}\mu_W(dx_j),
\endalign
$$
where the functions $f_V(x_j\,j\in V)$, $V\subset\{1,\dots,k\}$,
are those functions defined in formula (9.2) which appear in the
Hoeffding decomposition of the function $f(x_1,\dots,x_k)$, the
constants $C(k,V)$ are the limits appearing in the limit relation
$\lim\limits_{n\to\infty}C(n,k,V)=C(k,V)$ satisfied by the
coefficients $C(n,k,V)$ in formula (9.9), and $\mu_W$ is a white
noise with reference measure~$\mu$.}

\medskip
An essential step of the proof of Theorem 10.4 was the reduction
of the case of general kernel functions to the case of elementary
kernel functions. Let me make some comments about it.

It would be simple to make such a reduction if we had a good 
approximation of a canonical function with such elementary 
functions which are also canonical. But it is very hard to find
such an approximation. To overcome this difficulty we reduced the 
proof of Theorem~10.4 to a modified version of this result, where 
instead of a limit theorem for degenerate $U$-statistics a limit 
theorem for the random variables $J'_{n,k}(f)$ introduced in 
formula~(C10) has to be proved. In the proof of such a version we 
could apply the approximation of a general kernel function with 
not necessarily canonical elementary functions. Theorem~9.4 
helped us to work with such an approximation. Another natural way 
to overcome the above difficulty is to apply a Poissonian 
approximation of the normalized empirical measure. Such an 
approach was applied in~[14] and in~[31], where some 
generalizations of Theorem~10.4 were proved.

\beginsection Appendix D. The proof of Theorem 14.3.

{\it A result about the comparison of $U$-statistics and decoupled
$U$-statistics.}

\medskip\noindent
{\it The proof of Theorem 14.3.}\/ It will be simpler to formulate
and prove a generalized version of Theorem~14.3 where such
generalized $U$-statistics are considered in which different kernel
functions may appear in each term of the sum. More explicitly, let
$\ell=\ell(n,k)$ denote the set of all such sequences
$l=(l_1,\dots,l_k)$ of integers of length~$k$ for which
$1\le l_j\le n$, $1\le j\le k$. To define generalized $U$-statistics let 
us fix a set of functions
$\{f_{l_1,\dots,l_k}(x_1,\dots,x_k),\;(l_1,\dots,l_k)\in\ell\}$
which map the space $(X^k,\Cal X^k)$ to a separable Banach
space~$B$, and have the property $f_{l_1,\dots,l_k}(x_1,\dots,x_k)\equiv0$ 
if $l_j=l_{j'}$ for some indices $j\neq j'$. (The last condition corresponds 
to that property of $U$-statistics that the diagonals are omitted from 
the summation in their definition.) Let us denote this set of functions by 
$f(\ell)$ and define, similarly to the $U$-statistics and decoupled
$U$-statistics the generalized $U$-statistics and generalized
decoupled $U$-statistics by the formulas
$$
I_{n,k}(f(\ell))=\frac1{k!}\sum_{(l_1,\dots,l_k)\colon \;
1\le l_j\le n,\;j=1,\dots,k}
f_{l_1,\dots,l_k}\(\xi_{l_1},\dots,\xi_{l_k}\) \tag D1
$$
and
$$
\bar I_{n,k}(f(\ell))=\frac1{k!}\sum_{ (l_1,\dots,l_k)\colon \;
1\le l_j\le n,\; j=1,\dots,k}
f_{l_1,\dots,l_k}\(\xi_{l_1}^{(1)},\dots,\xi_{l_k}^{(k)}\) \tag D2
$$
(with the same independent random variables $\xi_{l}$ and
$\xi_{l}^{(j)}$, $1\le l\le n$, $1\le j\le k$, as in the definition
of the original $U$-statistics and decoupled $U$-statistics.)

The following generalization of relation (14.13) will be proved.
$$
P\(\|I_{n,k}(f(\ell))\|>u\)\le A(k)P\(\|\bar I_{n,k}(f(\ell))\|>\gamma(k)u\)
\tag14.13d
$$
with some constants $A(k)>0$ and $\gamma(k)>0$ depending only 
on the order~$k$ of these generalized $U$-statistics.

We concentrate mainly on the proof of the generalization (14.13d) of
relation (14.13). Formula~(14.14) is a relatively simple
consequence of it. Formula~(14.13d) will be proved by means of an 
inductive procedure which works only in this more general setting. 
It will be derived from the following statement.

Let us take two independent copies $\xi_{1}^{(1)},\dots,\xi_n^{(1)}$
and $\xi_{1}^{(2)},\dots,\xi_n^{(2)}$ of our original sequence of
random variables $\xi_1,\dots,\xi_n$, and introduce for all sets
$V\subset \{1,\dots,k\}$ the function $\alpha_V(j)$, $1\le j\le k$,
defined as $\alpha_V(j)=1$ if $j\in V$ and $\alpha_V(j)=2$ if
$j\notin V$. Let us define with their help the following
version of decoupled $U$-statistics
$$
\align
I_{n,k,V}(f(\ell))&=\frac1{k!}\sum_{ (l_1,\dots,l_k)\colon \;
1\le l_j\le n,\; j=1,\dots,k}
\!\!\!\!
f_{l_1,\dots,l_k}
\(\xi_{l_1}^{(\alpha_V(1))},\dots,\xi_{l_k}^{(\alpha_V(k))}\)\\
&\qquad\qquad\qquad\qquad \text{for all }V\subset \{1,\dots,k\}. \tag D3
\endalign
$$

The following inequality will be proved: There are some constants
$C_k>0$ and $D_k>0$ depending only on the order $k$ of the
generalized $U$-statistic $I_{n,k}(f(\ell))$ such that for all
numbers $u>0$
$$
P\(\|I_{n,k}(f(\ell))\|>u\)\le
\sum_{V\subset\{1,\dots,k\},\,1\le|V|\le k-1} C_kP\(D_k\|
I_{n,k,V}(f(\ell))\|>u\). \tag D4
$$
Here $|V|$ denotes the cardinality of the set $V$, and the condition
$1\le |V|\le k-1$ in the summation of formula (D4) means that the
sets $V=\emptyset$ and $V=\{1,\dots,k\}$ are omitted from the
summation, i.e. the terms where either $\alpha_V(j)=1$ 
or $\alpha_V(j)=2$ for all $1\le j\le k$ are not considered.
Formula (14.13d) can be derived from formula~(D4) by means 
of an inductive argument. The hard part of the 
problem is to prove formula~(D4). To do this first the following 
simple lemma will be proved.

\medskip\noindent
{\bf Lemma D1.} {\it Let $\xi$ and $\eta$ be two independent and
identically distributed random variables taking values in a
separable Banach space~$B$. Then
$$
3P\(|\xi+\eta|>\frac 23u\)\ge P(|\xi|>u)\quad \text{for all }u>0.
$$
}

\medskip\noindent
{\it Proof of Lemma D1.}\/ {\it Let $\xi$, $\eta$ and $\zeta$ be
three independent, identically distributed random variables taking
values in~$B$. Then
$$
\align
3P\(|\xi+\eta|>\frac23 u\)&=P\(|\xi+\eta|>\frac23 u\)+
P\(|\xi+\zeta|>\frac23 u\)+P\(|-(\eta+\zeta)|>\frac23 u\)\\
&\ge P(|\xi+\eta+\xi+\zeta-\eta-\zeta|>2u)=P(|\xi|>u).
\endalign
$$
}

\medskip
To prove formula (D4) we introduce the random variable
$$
T_{n,k}(f(\ell))=\frac1{k!}
\summ\Sb (l_1,\dots,l_k),\; (s_1,\dots,s_k) \colon \\
1\le l_j\le n,\, s_j=1 \text{ or }s_j=2,\; j=1,\dots, k,\endSb
\!\!\!\!\!\!\!\!\!\!\!
f_{l_1,\dots,l_k}\(\xi_{l_1}^{(s_1)},\dots,\xi_{l_k}^{(s_k)}\)
= \!\!\! \sum_{V\subset\{1,\dots,k\}}\!\!\!\!\!
I_{n,k,V}(f(\ell)). \tag D5
$$
Observe that the random variables $I_{n,k}(f(\ell))$,
$I_{n,k,\emptyset}(f(\ell))$ and $I_{n,k,\{1,\dots,k\}}(f(\ell))$
are identically distributed, and the last two random variables are
independent of each other. Hence Lemma~D1 yields that
$$ \allowdisplaybreaks
\align
P(\|I_{n,k}(f(\ell))\|>u)
&\le3P\(\|I_{n,k,\emptyset}(f(\ell))
+I_{n,k,\{1,\dots,k\}}(f(\ell))\|>\frac23u\)\\
&=3P\(\left\|T_{n,k}(f(\ell))-\!\!\!\!\!\!
\sum_{V\colon\; V\subset\{1,\dots,k\},\,
1\le|V|\le k-1} I_{n,k,|V|}(f(\ell))\right\|>\frac23u\) \!\!\!\!\!\!
\\
&\le 3P(3\cdot2^{k-1}\|T_{n,k}(f(\ell))\|>u) \tag D6    \\
&\qquad+
\!\!\!\!\!\!\!\!\!
\summ_{V\colon\; V\subset\{1,\dots,k\},\, 1\le|V|\le k-1}
\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!
3P(3\cdot2^{k-1}\|I_{n,k,|V|}(f(\ell))\|>u).
\endalign
$$
To derive relation (D4) from relation (D6) a good estimate is needed
on the probability $P(3\cdot2^{k-1}\|T_{n,k}(f(\ell))\|>u)$. To get
such an estimate the tail distribution of $\|T_{n,k}(f(\ell))\|$
will be compared with that of $\|I_{n,k,V}(f(\ell))\|$ for an
arbitrary set $V\subset\{1,\dots,k\}$. This will be done with the
help of Lemmas~D2 and~D4 formulated below.

In Lemma~D2 such a random variable $\|\bar I_{n,k,V}(f(\ell))\|$
will be constructed whose distribution agrees with that of
$\|I_{n,k,V}(f(\ell))\|$. The expression $\bar I_{n,k,V}(f(\ell))$,
whose norm will be investigated will be defined in formulas~(D7) 
and~(D8). It is a random polynomial of some Rademacher functions 
$\e_1,\dots,\e_n$. The coefficients of this polynomial are random 
variables, independent of the Rademacher functions 
$\e_1,\dots,\e_n$. Beside this, the constant term of this 
polynomial equals $T_{n,k}(f(\ell)$. These properties of the 
polynomial $\bar I_{n,k,V}(f(\ell))$ together with Lemma~D4 
formulated below enable us prove such an estimate on the 
distribution of $\|T_{n,k}(f(\ell))\|$ that together with 
formula~(D6) imply relation~(D4). Let us formulate these lemmas.

\medskip\noindent
{\bf Lemma D2.} {\it Let us consider a sequence of independent
random variables $\e_1,\dots,\e_n$, $P(\e_l=1)=P(\e_l=-1)=\frac12$,
$1\le l\le n$, which is also independent of the random variables
$\xi_{1}^{(1)},\dots,\xi_n^{(1)}$ and
$\xi_{1}^{(2)},\dots,\xi_n^{(2)}$ appearing in the definition of
the modified decoupled $U$-statistics $I_{n,k,V}(f(\ell))$ given
in formula (D3). Let us define with their help the sequences of
random variables $\eta_{1}^{(1)},\dots,\eta_n^{(1)}$ and
$\eta_{1}^{(2)},\dots,\eta_n^{(2)}$ whose elements
$(\eta_l^{(1)},\eta_l^{(2)})=(\eta_l^{(1)}(\e_l),\eta_l^{(2)}(\e_l))$,
$1\le l\le n$, are defined by the formula
$$
(\eta_l^{(1)}(\e_l),\eta_l^{(2)}(\e_l))=\(\frac{1+\e_l}2\xi_l^{(1)}+
\frac{1-\e_l}2\xi_l^{(2)},\frac{1-\e_l}2\xi_l^{(1)}+
\frac{1+\e_l}2\xi_l^{(2)}\),
$$
i.e. let $(\eta_l^{(1)}(\e_l),\eta_l^{(2)}(\e_l))=
(\xi_l^{(1)},\xi_l^{(2)})$ if $\e_l=1$, and
$(\eta_l^{(1)}(\e_l),\eta_l^{(2)}(\e_l))=
(\xi_l^{(2)},\xi_l^{(1)})$ if $\e_l=-1$, $1\le l\le n$.
Then the joint distribution of the pair of sequences of random
variables $\xi_{1}^{(1)},\dots,\xi_n^{(1)}$ and
$\xi_{1}^{(2)},\dots,\xi_n^{(2)}$ agrees with that of the pair of
sequences $\eta_{1}^{(1)},\dots,\eta_n^{(1)}$ and
$\eta_{1}^{(2)},\dots,\eta_n^{(2)}$, which is also independent of the
sequence $\e_1,\dots,\e_n$. 

Let us fix some $V\subset\{1,\dots,k\}$, and introduce the random
variable
$$
\bar I_{n,k,V}(f(\ell))=\frac1{k!}\sum_{(l_1,\dots,l_k) \colon\;
1\le l_j\le n,\; j=1,\dots,k}
f_{l_1,\dots,l_k}\(\eta_{l_1}^{(\alpha_V(1))},\dots,
\eta_{l_k}^{(\alpha_V(k))}\), \tag D7
$$
where similarly to formula (D3) $\alpha_V(j)=1$ if $j\in V$, and
$\alpha_V(j)=2$ if $j\notin V$. Then the identity
$$
\align
&2^k\bar I_{n,k,V}(f(\ell))  \tag D8 \\
&\qquad=\frac1{k!}\summ\Sb (l_1,\dots,l_k), \;(s_1,\dots,s_k)\colon \\
1\le l_j\le n,\; s_j=1 \text{ or }s_j=2,\;j=1,\dots, k, \endSb
\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!
(1+\kappa^{(1)}_{s_1,V}\e_{l_1})\cdots (1+\kappa^{(k)}_{s_k,V}\e_{l_k})
f_{l_1,\dots,l_k}\(\xi_{l_1}^{(s_1)},\dots,\xi_{l_k}^{(s_k)}\)
\endalign
$$
holds, where $\kappa^{(j)}_{1,V}=1$ and $\kappa^{(j)}_{2,V}=-1$ if
$j\in V$, and $\kappa^{(j)}_{1,V}=-1$ and $\kappa^{(j)}_{2,V}=1$ if
$j\notin V$, i.e. $\kappa_{1,V}^{(j)}=3-2\alpha_V(j)$ and 
$\kappa_{2,V}^{(j)}=-\kappa_{1,V}^{(j)}$.}

\medskip
Before the formulation of Lemma~D4 another Lemma~D3 will be
presented which will be applied in its proof.

\medskip\noindent
{\bf Lemma D3.} {\it Let $Z$ be a random variable taking values in 
a separable Banach space $B$ with expectation zero, i.e. let 
$E\kappa(Z)=0$ for all $\kappa\in B'$, where $B'$ denotes the 
(Banach) space of all (bounded) linear transformations of $B$ to 
the real line. Then $P(\|v+Z\|\ge\|v\|)\ge \inff_{\kappa\in B'}
\frac{(E|\kappa(Z)|)^2}{4E\kappa(Z)^2}$ for all $v\in B$.}

\medskip\noindent
{\bf Lemma D4.} {\it Let us consider a positive integer $n$ and
a sequence of independent random variables  $\e_1,\dots,\e_n$,
$P(\e_l=1)=P(\e_l=-1)=\frac12$, $1\le l\le n$. Beside this,
fix some positive integer $k$, take a separable Banach space~$B$ and
choose some elements $a_s(l_1,\dots,l_s)$ of this Banach space $B$,
$1\le s\le k$, $1\le l_j\le n$, $l_j\neq l_{j'}$ if $j\neq j'$,
$1\le j,j'\le s$. With the above notations the inequality
$$
P\(\left\|v+\sum_{s=1}^k\sum \Sb (l_1,\dots,l_s)\colon \;
1\le l_j\le n,\; j=1,\dots, s,\\
l_j\neq l_{j'} \text{ if } j\neq j'\endSb
a_s(l_1,\dots,l_s)\e_{l_1}\cdots\e_{l_s}\right\|\ge\|v\|\)\ge c_k \tag D9
$$
holds for all $v\in B$ with some constant $c_k>0$ which depends
only on the parameter $k$.}

\medskip\noindent
{\it Proof of Lemma D2.}\/ Let us consider the conditional
joint distribution of the sequences of random variables
$\eta_{1}^{(1)},\dots,\eta_n^{(1)}$ and
$\eta_{1}^{(2)},\dots,\eta_n^{(2)}$ under the condition that the
random vector $\e_1,\dots,\e_n$ takes the value of some prescribed
$\pm1$ series of length~$n$. Observe that this conditional
distribution agrees with the joint distribution of the sequences
$\xi_{1}^{(1)},\dots,\xi_n^{(1)}$ and
$\xi_{1}^{(2)},\dots,\xi_n^{(2)}$ for all possible conditions.
This fact implies the statement about the joint distribution of
the sequences $\eta_l^{(1)},\eta_l^{(2)}$, $1\le l\le n$ and their
independence of the sequence $\e_1,\dots,\e_n$.

To prove identity~(D8) let us fix a set $M\subset\{1,\dots,n\}$,
and consider the case when $\e_l=1$ if $l\in M$ and $\e_l=-1$ if
$l\notin M$. Put $\beta_{V,M}(j,l)=1$ if $j\in V$ and $l\in M$ 
or $j\notin V$ and  $l \notin M$, and let $\beta_{V,M}(j,l)=2$ 
otherwise. Then we have for all $(l_1,\dots,l_k)$, $1\le l_j\le n$,
$1\le j\le n$, and our fixed set $V$
$$
\aligned
&\summ_{(s_1,\dots,s_k)\colon\; s_j=1 \text{ or }s_j=2,\;j=1,\dots, k,}
(1+\kappa^{(1)}_{s_1,V}\e_{l_1})\cdots (1+\kappa^{(k)}_{s_k,V}\e_{l_k})
f_{l_1,\dots,l_k}\(\xi_{l_1}^{(s_1)},\dots,\xi_{l_k}^{(s_k)}\)\\
&\qquad\qquad  \qquad=2^k f_{l_1,\dots,l_k}
\(\xi_{l_1}^{(\beta_{V,M}(1,l_1))},\dots,
\xi_{l_k}^{(\beta_{V,M}(k,l_k))}\),
\endaligned \tag D10
$$
since the product
$(1+\kappa^{(1)}_{s_1,V}\e_{l_1})\cdots (1+\kappa^{(k)}_{s_k,V}\e_{l_k})$
equals either zero or $2^k$, and it equals $2^k$ for that sequence
$(s_1,\dots,s_k)$ for which $\kappa^{(j)}_{s_j,V}\e_{l_j}=1$ for all 
$1\le j\le k$, and the relation $\kappa^{(j)}_{s_j,V}\e_{l_j}=1$ is 
equivalent to $\beta_{V,M}(j,l_j)=s_j$ for all $1\le j\le k$. (In 
relation~(D10) it is sufficient to consider only such products for 
which $l_j\neq l_{j'}$ if $j\neq j'$ because of the properties of 
the functions $f_{l_1,\dots,l_k}$.)

Beside this, $\xi_l^{\beta_{V,M}(l,j)}=\eta_l^{\alpha_V(j)}$ for all
$1\le l\le n$ and $1\le j\le k$, and as a consequence
$$f_{l_1,\dots,l_k}\(\xi_{l_1}^{(\beta_{V,M}(1,l_1))},\dots,
\xi_{l_k}^{(\beta_{V,M}(k,l_k))}\)=
f_{l_1,\dots,l_k}\(\eta_{l_1}^{(\alpha_V(1))},\dots,
\eta_{l_k}^{(\alpha_V(k))}\).
$$
Summing up the identities (D10) for all $1\le l_1,\dots,l_k\le n$
and applying the last
identity we get relation~(D8), since the identity obtained in such
a way holds for all $M\subset\{1,\dots,n\}$.

\medskip\noindent
{\it Proof of Lemma D3.}\/ Let us first observe that if $\xi$
is a real valued random variable with zero expectation, then
$P(\xi\ge0) \ge \frac{(E|\xi|)^2}{4E\xi^2}$ since $(E|\xi|)^2
=4(E(\xi I(\{\xi\ge0\}))^2\le 4P(\xi\ge0)E\xi^2$ by the Schwarz
inequality, where $I(A)$ denotes the indicator function of
the set~$A$. (In the above calculation and in the subsequent proofs
I apply the convention $\frac00=1$. We need this convention if
$E\xi^2=0$. In this case we have, because of the condition $E\xi=0$
the identity $P(\xi=0)=1$, hence the above proved identity holds in 
this case, too.)

Given some $v\in B$, let us choose a linear operator $\kappa$ such
that $\|\kappa\|=1$ and $\kappa(v)=\|v\|$. Such an operator exists
by the Banach--Hahn theorem. Observe that $\{\oo\colon\;\|v+Z(\oo)\|
\ge\|v\|\} \supset\; \{\oo\colon\; \kappa(v+Z(\oo))\ge\kappa(v)\}
=\{\oo\colon\; \kappa(Z(\oo))\ge0\}$. Beside this $E\kappa(Z)=0$.
Hence we can apply the above proved inequality for $\xi=\kappa(Z)$,
and it yields that $P(\|v+Z\|\ge\|v\|)\ge P(\kappa(Z)\ge0)
\ge\frac{(E|\kappa(Z)|)^2}{4E\kappa(Z)^2}$. Lemma~D3 is proved.

\medskip\noindent
{\it Proof of Lemma D4.}\/
Take the class of random polynomials
$$
Y=\sum_{s=1}^k\sum \Sb (l_1,\dots,l_s)\colon\;
1\le l_j\le n,\; j=1,\dots, s,\\
l_j\neq l_{j'} \text{ if } j\neq j'\endSb
b_s(l_1,\dots,l_s)\e_{l_1}\cdots\e_{l_s},
$$
where $\e_l$, $1\le l\le n$, are independent random variables with
$P(\e_l=1)=P(\e_l=-1)=\frac12$, and the coefficients
$b_s(l_1,\dots,l_s)$, $1\le s\le k$, are arbitrary real numbers.
The proof of Lemma~D4 can be reduced to the statement that there
exists a constant $c_k>0$ depending only on the order~$k$ of these
polynomials such that the inequality
$$
(E|Y|)^2\ge 4c_k EY^2. \tag D11
$$
holds for all such polynomials~$Y$. Indeed, consider the polynomial
$$
Z=\sum_{s=1}^k\sum \Sb (l_1,\dots,l_s)\colon\;
1\le l_j\le n,\; j=1,\dots, s,\\
l_j\neq l_{j'} \text{ if } j\neq j'\endSb
a_s(l_1,\dots,l_s)\e_{l_1}\cdots\e_{l_s},
$$
and observe that $E\kappa(Z)=0$ for all linear functionals $\kappa$
on the space $B$. Hence Lemma~D3 implies that the left-hand side
expression in~(D9) is bounded from below by
$\inff_{\kappa\in B'}\frac{(E|\kappa(Z)|)^2}{4E\kappa(Z)^2}$. On the
other hand, relation~(D11) implies that
$\inff_{\kappa\in G'}\frac{(E|\kappa(Z)|)^2}{4E\kappa(Z)^2}\ge c_k$.

To prove relation (D11) first we compare the moments $EY^2$ and
$EY^4$. Let us introduce the random variables
$$
Y_s=\sum \Sb (l_1,\dots,l_s)\colon \;
1\le l_j\le n,\; j=1,\dots, s,\\ l_j\neq l_{j'} \text{ if }
j\neq j'\endSb b_s(l_1,\dots,l_s)\e_{l_1}\cdots\e_{l_s}
\quad 1\le s\le k.
$$
We shall show that the estimates of Section~13 imply that
$$
EY_s^4\le 2^{4s} \(EY_s^2\)^2 \tag D12
$$
for these random variables $Y_s$.

Relation (D12) together with the uncorrelatedness of the random
variables $Y_s$, $1\le s\le k$, imply that
$$
\align
EY^4&=E\(\sum_{s=1}^k Y_s\)^4\le k^3\sum_{s=1}^k EY_s^4\le
k^3 2^{4k} \sum_{s=1}^k  (EY_s^2)^2\\
&\le k^3 2^{4k}\(\sum_{s=1}^k EY_s^2\)^2=k^3 2^{4k}(EY^2)^2.
\endalign
$$
This estimate together with the H\"older inequality with $p=3$ and 
$q=\frac32$ yield that $EY^2=E|Y|^{4/3}|\cdot|Y|^{2/3}\le 
(EY^4)^{1/3}(E|Y|)^{2/3}\le k2^{4k/3}(EY^2)^{2/3}(E|Y|)^{2/3}$, 
i.e. $EY^2\le k^32^{4k}(E|Y|)^2$, and relation (D11) holds with
$4c_k=k^{-3}2^{-4k}$. Hence to complete the proof of Lemma~D4
it is enough to check relation~(D12).

In the proof of relation (D12) it can be assumed that the
coefficients $b_s(l_1,\dots,l_s)$ of the random variable $Y_s$ are
symmetric functions of the arguments
$l_1,\dots,l_s$, since a symmetrization of these coefficients does
not change the value of $Y$. Put
$$
B^2_s=\sum \Sb (l_1,\dots,l_s)\colon\;
1\le l_j\le n,\; j=1,\dots, s,\\
l_j\neq l_{j'} \text{ if } j\neq j'\endSb b_s^2(l_1,\dots,l_s),
\quad 1\le s\le k.
$$
Then
$$
EY_s^2=s! B_s^2,
$$
and
$$
EY_s^4\le 1\cdot3\cdot5\cdots(4s-1)B_s^4
=\frac{(4s)!}{2^{2s}(2s)!}B_s^4
$$
by Lemmas 13.4 and 13.5 with the choice $M=2$ and $k=s$. 
Inequality~(D12) follows from the last two relations. Indeed, to 
prove formula~(D12) by means of these relations it is enough to 
check that $\frac{(4s)!}{2^{2s}(2s)!(s!)^2}\le 2^{4s}$. But it is 
easy to check this inequality with induction with respect to $s$.  
(Actually, there is a well-known inequality in the literature, 
known under the name Borell's inequality, which implies 
inequality~(D12) with a better coefficient at the right hand side 
of this estimate.)

\medskip
Let us turn back to the estimation of the probability
$P(3\cdot2^{k-1}\|T_{n,k}(f)\|>u)$. Let us introduce the
$\sigma$-algebra $\Cal F=\Cal B(\xi_l^{(1)},\xi_l^{(2)},\,1\le
l\le n)$ generated by the random variables $\xi_l^{(1)},\xi_l^{(2)}$,
$1\le l\le n$, and fix some set $V\subset\{1,\dots,k\}$.
I show with the help of Lemma~D4 and formula~(D8) that there exists 
some constant $c_k>0$ such that the random
variables $T_{n,k}f(\ell))$ defined in formula~(D5) and 
$\bar I_{n,k,V}(f(\ell))$ defined in formula~(D7) satisfy
the inequality
$$
P\(\|2^k\bar I_{n,k,V}(f(\ell))\|>\|T_{n,k}(f(\ell))\||\Cal F\)
\ge c_k \quad \text{ with probability 1.} \tag D13
$$

In the proof I shall exploit that in formula~(D8) 
$2^k\bar I_{n,k,V}(f(\ell))$ is represented by a polynomial of the
Rademacher functions $\e_1,\dots,\e_n$ whose constant term is
$T_{n,k}(f(\ell))$. The coefficients of this polynomial are 
functions of the random variables $\xi^{(1)}_l$ and $\xi^{(2)}_l$, 
$1\le l\le n$.  The independence of these random variables from
$\e_{l}$, $1\le l\le n$, and the definition of the $\sigma$-algebra
$\Cal F$ yield that 
$$
\align
&P\(\|2^k\bar I_{n,k,V}(f(\ell))\|>\|T_{n,k}(f(\ell))\||\Cal F\)\\
&\qquad=P_{\e_V}\biggl(\biggl\|\frac1{k!} \!\!\!\!\!\!\!\!\!\!\!
\summ\Sb (l_1,\dots,l_k),\; (s_1,\dots,s_k)\colon\\
1\le l_j\le n, s_j=1 \text{ or }s_j=2,\; j=1,\dots, k,\endSb
\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!
(1+\kappa^{(1)}_{s_1,V}\e_{l_1})\cdots (1+\kappa^{(k)}_{s_k,V}\e_{l_k})
f_{l_1,\dots,l_k}\(\xi_{l_1}^{(s_1)},\dots,\xi_{l_k}^{(s_k)}\)\biggr\|\\
&\qquad \qquad\qquad\qquad\qquad\qquad
>\|T_{n,k}(f(\ell))(\xi_l^{(j)},\,1\le l\le n,\, j=1,2)\|\biggr),
\tag D14
\endalign
$$
where $P_{\e_V}$ means that  the values of the  random variables
$\xi_l^{(1)}$, $\xi_l^{(2)}$, $1\le l\le n$, are fixed, (their value
depend on the atom of the $\sigma$-algebra $\Cal F$ we are considering) 
and the probability is taken with respect to the remaining random variables
$\e_l$, $1\le l\le n$. At the right-hand side of (D14) the probability 
of such an event is considered that the norm of a polynomial of 
order~$k$ of the random variables $\e_1,\dots,\e_n$ is larger than
$\|T_{n,k}(f(\ell))(\xi_l^{(j)},\,1\le l\le n,\, j=1,2)\|$.
Beside this, the constant term of this polynomial
equals~$T_{n,k}(f(\ell))(\xi_l^{(j)},\,1\le l\le n,\, j=1,2)$.
Hence this probability can be bounded by
means of Lemma~D4, and this result yields relation~(D13).

As the distributions of $I_{n,k,V}(f(\ell))$ and
$\bar I_{n,k,V}(f(\ell))$ agree by the first statement of Lemma~D2 
and a comparison of formulas~(D3) and~(D7), relation (D13) implies that
$$
\align
&P\(\|2^k I_{n,k,V}(f(\ell))\|\ge\frac13\cdot2^{1-k} u\) 
=P\(\|2^k\bar I_{n,k,V}(f(\ell))\|\ge\frac13\cdot2^{1-k} u\) \\
&\qquad \ge P\(\|2^k\bar I_{n,k,V}(f(\ell))\|\ge\|T_{n,k}(f(\ell))\|,\;
\|T_{n,k}(f(\ell))\|\ge\frac13\cdot2^{1-k} u\)\\
&\qquad=\int_{\{\oo\colon\; \|T_{n,k}(f(\ell))(\oo)\|
\ge\frac13\cdot2^{1-k} u\}}
P\(\|2^k\bar I_{n,k,V}(f(\ell))\|>\|T_{n,k}(f(\ell))\||\Cal F\)\,dP\\
&\qquad \ge c_k P(3\cdot2^{k-1} \|T_{n,k}(f(\ell))\|\ge u).
\endalign
$$
The last inequality with the choice of any set $V\subset\{1,\dots,k\}$,
$1\le |V|\le k-1$, together with relation~(D6) imply formula~(D4).

Relation (14.13d) will be proved together with another inductive 
hypothesis with the help of relation~(D4) by means of an induction
procedure with respect to the order $k$ of the $U$-statistic. To 
formulate the other inductive hypothesis some new quantities will 
be introduced. Let $\Cal W=\Cal W(k)$
denote the set of all partitions of the set $\{1,\dots,k\}$. Let
us fix $k$ independent copies $\xi_{1}^{(j)},\dots,\xi_n^{(j)}$,
$1\le j\le k$, of the sequence of random variables
$\xi_{1},\dots,\xi_n$. Given a partition
$W=(U_1,\dots,U_s)\in\Cal W(k)$ let us introduce the function
$s_W(j)$, $1\le j\le k$, which tells for all arguments $j$ the index
of that element of the partition~$W$ which contains the point $j$,
i.e. the value of the function $s_W(j)$, $1\le j\le k$, in a point $j$ 
is defined by the relation $j\in V_{s_W(j)}$. Let us introduce the 
expression
$$
\align
I_{n,k,W}(f(\ell))&=\frac1{k!}\sum_{ (l_1,\dots,l_k)\colon \;
 1\le l_j\le n,\;j=1,\dots,k}
f_{l_1,\dots,l_k}\(\xi_{l_1}^{(s_W(1))},\dots,\xi_{l_k}^{(s_W(k))}\)\\
&\qquad\qquad\qquad\qquad \text{for all }W\in\Cal W(k).
\endalign
$$
An expression of the  form $I_{n,k,W}(f(\ell))$, $W\in\Cal W_k$, 
will be called a decoupled $U$-statistic with generalized 
decoupling. Given a partition $W=(U_1,\dots,U_s)\in\Cal W_k$ 
let us call the number $s$, i.e.\ the number of the elements of 
this partition the rank both of the partition $W$ and of the 
decoupled $U$-statistic $I_{n,k,W}(f(\ell))$ with generalized 
decoupling.

Now I formulate the following hypothesis. For all $k\ge2$ and 
$2\le j\le k$ there  exist some constants $C(k,j)>0$ and $\delta(k,j)>0$ 
such that for all $W\in\Cal W_k$ a decoupled $U$-statistic 
$I_{n,k,W}(f(\ell))$ with generalized decoupling satisfies the inequality
$$
\aligned
&P(\|I_{n,k,W}(f(\ell))\|>u)\le C(k,j)P\(\|\bar
I_{n,k}(f(\ell))\|>\delta(k,j) u\) \\
&\qquad\qquad\qquad\text{for all }2\le j\le k \text{ if the rank of } W
\text{ equals }j.
\endaligned \tag D15
$$

It will be proved by induction with respect to $k$ that both 
relations~(14.13d) and~(D15) hold for $U$-statistics of order~$k$.
Let us observe that for $k=2$ relation~(14.13d) follows from~(D4).
Relation~(D15) also holds for $k=2$, since in this case we have to 
consider only the case $j=k=2$, and relation (D15) clearly holds 
in this case with $C(2,2)=1$ and
$\delta(2,2)=1$. Hence we can start our inductive proof with $k=3$.
First I prove relation~(D15).

In relation (D15) the tail-distribution of decoupled
$U$-sta\-tis\-tics with generalized decoupling is compared with that
of the decoupled $U$-statistic $\bar I_{n,k}(f(\ell))$ introduced
in~(D2). Given the order $k$ of these $U$-statistics it will be proved 
by means of a backward induction with
respect to the rank $j$ of the decoupled $U$-statistics
$I_{n,k,W}(f(\ell))$ with generalized decoupling.

Relation (D15) clearly holds for $j=k$ with $C(k,k)=1$ and
$\delta(k,k)=1$. To prove it for decoupled $U$-statistics 
with generalized decoupling of rank $2\le j<k$ first the following
observation will be made. If the rank~$j$ of the partition
$W=(U_1,\dots,U_j)$ satisfies the relation $2\le j\le k-1$, then
it contains an element with cardinality strictly less than $k$ and
strictly greater than 1. For the sake of simpler notation let us
assume that the element $U_j$ of this partition is such an element,
and $U_j=\{t,\dots,k\}$ with some $2\le t\le k-1$. The
investigation of general $U$-statistics of rank $j$,
$2\le j\le k-1$, can be reduced to this case by a
reindexation of the arguments in the $U$-statistics if it is
necessary. Let us consider the partition $\bar W=(U_1,\dots,
U_{j-1},\{t\},\dots,\{k\})$ and the decoupled $U$-statistic
$I_{n,k,\bar W}(f(\ell))$ with generalized decoupling corresponding to
this partition~$\bar W$. It will be shown that our inductive
hypothesis implies the inequality
$$
P(\|I_{n,k,W}(f(\ell))\|>u)\le \bar A(k) P\(\|I_{n,k,\bar W}
(f(\ell))\|>\bar \gamma(k) u\) \tag D16
$$
with $\bar A(k)=\supp_{2\le p\le k-1}A(p)$,
$\bar\gamma(k)=\inff_{2\le p\le k-1}\gamma(p)$ if the 
rank $j$ of $W$ is such that $2\le j\le k-1$, where the constants 
$A(p)$ and $\gamma(p)$ agree with the corresponding coefficients in 
formula~(14.13d).

To prove relation~(D16) (in the case when $U_j=\{t,\dots,k\}$) let 
us define the $\sigma$-algebra $\Cal F$ generated by the random 
variables appearing in the first $t-1$ coordinates of these 
$U$-statistics, i.e. by the random variables $\xi^{s_W(j)}_{l_j}$,
$1\le j\le t-1$, and $1\le l_j\le n$ for all $1\le j\le t-1$. We have
$2\le t\le k-1$. By our inductive hypothesis relation~(14.13d) holds 
for $U$-statistics of order $p=k-t+1$, since $2\le p\le k-1$. I 
claim that this implies that
$$
P(\|I_{n,k,W}(f(\ell))\|>u|\Cal F)\le A(k-t+1)
P\(\|I_{n,k,\bar W}(f(\ell))\|>\gamma(k-t+1)u|\Cal F\) \tag D17
$$ 
with probability~1. Indeed, by the independence properties of 
the random variables $\xi_l^{s_W(j)}$ 
(and $\xi_l^{s_{\bar W}(j)}$), 
$1\le j\le k$, $1\le l\le n$,
$$
P(\|I_{n,k,W}(f(\ell))\|>u|\Cal F)
=P_{\xi_l^{s_W(j)},1\le j\le t-1}(\|I_{n,k,W}(f(\ell)\|>u)
$$
and
$$
P\(\|I_{n,k,\bar W}(f(\ell))\|>\gamma(k-t+1)u|\Cal F\)
=P_{\xi_l^{s_W(j)},1\le j\le t-1}(\|I_{n,k,\bar W}f(\ell)\|
>\gamma(k-t+1)u),
$$ 
where $P_{\xi_l^{s_W(j)}, 1\le j\le t-1}$ denotes that the values of the 
random variables $\xi_l^{s_W(j)}(\oo)$, $1\le j\le t-1$, $1\le l\le n$, 
are fixed, and we consider the probability that the appropriate
functions of these fixed values and of the remaining random variables 
$\xi^{s_W(j)}$ and $\xi^{s_{\bar W}(j)}$, $t\le j\le k$,
satisfy the desired relation. These identities and the relation 
between the sets $W$ and $\bar W$ imply that relation (D17) is 
equivalent  to the identity (14.13d) for the generalized 
$U$-statistics of order $2\le k-t+1\le k-1$ with kernel functions
$$
\align
&f_{l_t,\dots,l_k}(x_t,\dots,x_k)\\
&\qquad=\sum_{(l_1,\dots,l_{t-1})\colon\;1\le l_j\le n,\;1\le j\le t-1}
f_{l_1,\dots,l_k}(\xi_{l_1}^{s_W(1)}(\oo),\dots,
\xi_{l_{t-1}}^{s_W(t-1)}(\oo),x_t,\dots,x_k).
\endalign
$$
Relation~(D16) follows from inequality~(D17) if expectation is taken 
at both sides. As the rank of $\bar W$ is strictly greater than the 
rank of $W$, relation~(D16) together with our backward inductive 
assumption imply relation~(D15) for all $2\le j\le k$.

Relation~(D15) implies in particular (with the applications of 
partitions of order~$k$ and rank~2) that the terms in the sum at the
right-hand side of~(D4) satisfy the inequality
$P\(D_k\|I_{n,k,V}(f(\ell))\|>u\)\le \bar C(k,j)
P\(\|\bar I_{n,k}(f(\ell))\|>\bar D_k u\)$ with some appropriate
$\bar C_k>0$ and $\bar D_k>0$ for all $V\subset\{1,\dots,k\}$, 
$1\le|V|\le k-1$. This inequality together with relation~(D4)
imply that inequality~(14.13d) also holds for the parameter~$k$. 

\medskip
In such a way we get the proof of relation (14.13d) and of its
special case, relation~(14.13). Let us prove
formula~(14.14) with its help first in the simpler case when the
supremum of finitely many functions is taken. If $M<\infty$
functions $f_1,\dots,f_M$ are considered, then relation (14.14) 
for the supremum of the $U$-statistics and decoupled 
$U$-statistics with these kernel functions can be derived from 
formula (14.13) if it is applied for the function 
$f=(f_1,\dots,f_M)$ with values in the separable Banach space 
$B_M$ which consists of the vectors 
$(v_1,\dots,v_M)$, $v_j\in B$, $1\le j\le M$, and the norm
$\|(v_1,\dots,v_M)\|=\supp_{1\le j\le m}\|v_j\|$ is introduced in
it. The application of formula (14.13) with this choice yields
formula~(14.14) for this supremum. Let us emphasize that the
constants appearing in this estimate do not depend on the
number~$M$. (We took only $M<\infty$ kernel functions, because
with such a choice the Banach space $B_M$ defined above
is also separable.) Since the distribution of the random 
variables $\supp_{1\le s\le M}\left\|I_{n,k}(f_s)\right\|$ 
converge to that of 
$\supp_{1\le s<\infty} \left\| I_{n,k}(f_s)\right\|$, and
the distribution of the random variables $\supp_{1\le s\le M}
\left\| \bar I_{n,k}(f_s)\right\|$ converge to that of
$\supp_{1\le s<\infty}\left\|\bar I_{n,k}(f_s)\right\|$ as
$M\to\infty$, relation (14.14) in the general case follows from its
already proved special case and a limiting procedure $M\to\infty$.

\medskip\noindent
{\it Remark.} The above proved formula (14.13d) can be slightly 
generalized. It also holds if the expressions $I_{n,k}(f(\ell))$ 
and $\bar I_{n,k}(f(\ell))$ appearing in this inequality are defined
in a more general way. Namely, they are the random functions 
introduced in formulas (D1) and (D2), but the sequences
$\xi_1,\dots,\xi_n$ and their independent copies
$\xi_1^{(j)},\dots,\xi_n^{(j)}$ in these formulas are independent
random variables which may also be non-identically distributed.
This generalization can be shown without any essential change in
the original proof.

\parskip=1pt plus 0.5pt

\beginsection References:

\item{1.)} Adamczak, R. (2006) Moment inequalities for
$U$-statistics. {\it Annals of Probability} {\bf34}, 2288--2314
\item{2.)} Alexander, K. (1987) The central limit theorem for
empirical processes over Vapnik--\v{C}ervonenkis classes. {\it Annals
of Probability} {\bf 15}, 178--203
\item{3.)} Arcones, M. A. and Gin\'e, E. (1993) Limit theorems for
$U$-processes. {\it Annals of Probability}, {\bf 21}, 1494--1542
\item{4.)} Arcones, M. A. and Gin\'e, E. (1994) $U$-processes
indexed by Vapnik--\v{C}ervonenkis classes of functions with
application to asymptotics and bootstrap of $U$-statistics with
estimated parameters. {\it Stoch. Proc. Appl.}  {\bf 52}, 17--38
\item{5.)} Bennett, G. (1962) Probability inequality for the sum of
independent random variables. {\it J. Amer. Statist. Assoc.}\/
{\bf 57}, 33--45
\item{6.)} Bonami, A. (1970) \'Etude des coefficients de Fourier des
fonctions de $L^p(G)$. {\it Ann. Inst. Fourier (Grenoble)\/} {\bf 20}
335--402
\item{7.)} de la Pe\~na, V. H. and Gin\'e, E. (1999) {\it
Decoupling. From dependence to independence.}\/ Springer series in
statistics. Probability and its application. Springer Verlag,
New York, Berlin, Heidelberg
\item{8.)} de la Pe\~na, V. H. and  Montgomery--Smith, S. (1995)
Decoupling inequalities for the tail-probabilities of multivariate
$U$-statistics. {\it Ann. Probab.}, {\bf 23}, 806--816
\item{9.)} Dobrushin, R. L. (1979) Gaussian and their subordinated
fields.  {\it Annals of Probability}\/ {\bf 7}, 1-28
\item{10.)} Dudley, R. M. (1978) Central limit theorems for empirical
measures. {\it Annals of Probability}\/ {\bf 6}, 899--929
\item{11.)} Dudley, R. M. (1984) A course on empirical processes.
{\it Lecture Notes  in Mathematics}\/ {\bf 1097}, 1--142 Springer
Verlag, New York
\item{12.)} Dudley, R. M. (1989)  {\it Real Analysis and
Probability.}\/ Wadsworth \& Brooks, Pacific Grove, California
\item{13.)} Dudley, R. M. (1998)  {\it Uniform Central Limit
Theorems.}\/ Cambridge University Press, Cambridge U.K.
\item{14.)} Dynkin, E. B. and Mandelbaum, A. (1983) Symmetric
statistics, Poisson processes and multiple Wiener integrals. {\it
Annals of Statistics\/} {\bf 11}, 739--745
\item{15.)} Frankl, P. and Pach J. (1983) On the number of sets in
null-$t$-design. {\it European J.  Combinatorics} {\bf 4} 21--23
\item{16.)} Gin\'e, E. and Guillou, A. (2001) On consistency of
kernel density estimators for randomly censored data: Rates holding
uniformly over adaptive intervals. {\it Ann. Inst. Henri
Poincar\'e PR\/} {\bf 37} 503--522
\item{17.)} Gin\'e, E., Kwapie\'n, S, Lata\l{}a, R. and Zinn, J.
(2001) The LIL for canonical $U$-statistics of order~2.
{Annals of Probability} {\bf 29} 520--527
\item{18.)} Gin\'e, E., Lata\l{}a, R. and Zinn, J. (2000)
Exponential and moment inequalities for $U$-statistics in {\it High
dimensional probability II.} Progress in Probability 47. 13--38.
Birkh\"auser Boston, Boston, MA.
\item{19.)} Gross, L. (1975) Logarithmic Sobolev inequalities.
Amer. J. Math.  {\bf 97}, 1061--1083
\item{20.)} Guionnet, A. and Zegarlinski, B. (2003) Lectures on
Logarithmic Sobolev inequalities. {\it Lecture Notes in Mathematics}
{\bf 1801} 1--134 2. Springer Verlag, New York
\item{21.)} Hanson, D. L. and Wright, F. T. (1971) A bound on the
tail probabilities for quadratic forms  in independent random
variables. {\it Ann. Math. Statist.} {\bf 42} 52--61
\item{22.)} Hoeffding, W. (1948) A class of statistics with
asymptotically normal distribution. {\it Ann. Math. Statist.}
{\bf 19} 293--325
\item{23.)} Hoeffding, W. (1963) Probability inequalities for sums
of bounded random variables. {\it J. Amer. Math. Society}\/
{\bf 58}, 13--30
\item{24.)} It\^o K. (1951) Multiple Wiener integral. {\it J. Math.
Soc. Japan}\/  {\bf3}.  157--164
\item{25.)} Kaplan, E.L. and Meier P. (1958) Nonparametric
estimation from incomplete data, {\it Journal of American
Statistical Association}, {\bf 53}, 457--481.
\item{26.)} Lata\l{a}, R. (2006) Estimates of moments and tails of
Gaussian chaoses. {\it Annals of Probability} {\bf34} 2315--2331
\item{27.)} Ledoux, M. (1996) On Talagrand deviation inequalities
for product measures. {\it ESAIM: Probab. Statist.}\/ {\bf 1.}
63--87. Available at http://www.emath./fr/ps/.
\item{28.)} Ledoux, M. (2001) The concentration of measure phenomenon.
{\it Mathematical Surveys and Monographs}\/ {\bf 89} American Mathematical
Society, Providence, RI.
\item{29.)} Major, P. (1981) Multiple Wiener--It\^o integrals. {\it
Lecture Notes in Mathematics\/} {\bf 849}, Springer Verlag, Berlin,
Heidelberg, New York,
\item{30.)} Major, P. (1988) On the tail behaviour of the
distribution function of multiple stochastic integrals. {\it
Probability Theory and Related Fields}, {\bf 78},  419--435
\item{31.)} Major, P. (1994) Asymptotic distributions for weighted
$U$-statistics. {\it The Annals of Probability}, {\bf 22} 1514--1535
\item{32.)} Major, P. (2005) An estimate about multiple stochastic
integrals with respect to a normalized empirical measure.
{\it Studia Scientarum Mathematicarum Hungarica.} 295--341
\item{33.)} Major, P. (2005) Tail behaviour of multiple random
integrals and $U$-sta\-tis\-tics. {\it Probability Reviews.} 448--505
\item{34.)} Major, P. (2006) An estimate on the maximum of a nice
class of stochastic integrals. {\it Probability Theory
and Related Fields.} {\bf 134}, 489--537
\item{35.)} Major, P. (2006) A multivariate generalization of
Hoeffding's inequality. {\it Electronic Communication in
Probability} {\bf 2} (220--229)
\item{36.)} Major, P. (2007) On a multivariate version of
Bernstein's inequality {\it Electronic Journal of
Probability} {\bf12} 966--988
\item{37.)} Major, P. (2005) On the tail behaviour of multiple
random integrals and degenerate $U$-statistics. (First version of
this lecture note) http://www.renyi.hu/\~{}major
\item{38.)} Major, P. and Rejt\H{o}, L. (1988) Strong embedding of
the distribution function under random censorship. {\it Annals of
Statistics} {\bf 16}, 1113--1132
\item{39.)} Major, P. and Rejt\H{o}, L. (1998) A note on
nonparametric estimations. In the conference volume to the 65.
birthday of Mikl\'os Cs\"org\H{o}. 759--774
\item{40.)} Malyshev, V. A. and Minlos, R. A. (991) Gibbs Random
Fields. Method of cluster expansion. Kluwer, Academic Publishers,
Dordrecht
\item{41.)} Massart, P. (2000) About the constants in Talagrand's
concentration inequalities for empirical processes. {\it Annals of
Probability}\/ {\bf 28}, 863--884
\item{42.)} Mc. Kean, H. P. (1973) Wiener's theory of non-linear
noise. in {\it Stochastic Differential Equations}
SIAM--AMS Proc. 6 197--209
\item{43.)} Nelson, E. (1973) The free Markov field. J. Functional
Analysis {\bf 12}, 211--227
\item{44.)} Pollard, D. (1984) {\it Convergence of Stochastic
Processes.}\/ Springer Verlag, New York
\item{45.)} Rota, G.-C. and Wallstrom, C. (1997) Stochastic
integrals: a combinatorial approach. {\it Annals of Probability}
{\bf 25} (3) 1257--1283
\item{46.)} Surgailis, D. (1984) On multiple Poisson stochastic
integrals and associated Markov semigroups. {\it Probab. Math.
Statist.} 3. no. {\bf 2} 217-239
\item{47.)} Surgailis, D. (2000) Long-range dependence and Appell
rank. {\it Annals of Probability} {\bf 28} 478--497
%\item{41.)} Surgailis, D. (2000) CLTs for polynomials of linear
%sequences: Diagram formulae with illustrations. in {\it Long Range
%Dependence} 111--128 Birkh\"auser, Boston, Boston, MA.
\item{48.)} Szeg\H{o}, G. (1967) {\it Orthogonal Polynomials.} 
American Mathematical Society Colloquium Publications. Vol. 23
\item{49.)} Takemura, A. (1983) Tensor Analysis of ANOVA
decomposition. {\it J. Amer. Statist. Assoc.} {\bf 78}, 894--900
\item{50.)} Talagrand, M. (1994) Sharper bounds for Gaussian and
empirical processes. {\it Annals of Probability} {\bf 22}, 28--76
\item{51.)} Talagrand, M. (1996) New concentration inequalities in
product spaces. {\it Invent. Math.} {\bf 126}, 505--563
\item{52.)} Talagrand, M. (2005) {\it The general chaining.}
Springer Monographs in Mathematics. Springer Verlag, Berlin 
Heidelberg New York
\item{53.)} Vapnik, V. N. (1995) {\it The Nature of Statistical
Learning Theory.} Springer Verlag, New York

\vfill\eject

\centerline {\script CONTENT}
$$
\vbox{\halign{\hfill # \ &\vtop{\hsize=12truecm\parindent=0pt #
\vskip3pt} \quad &\vtop{\hsize=0.5truecm\parindent=0pt #
\vskip3pt} \cr
1. & Introduction. \dotfill &\rightline{1}\cr
2. & Motivation of the investigation. Discussion of some problems.
\dotfill & \rightline{3}\cr
3. & Some estimates about sums of independent random variables.
\dotfill & \rightline{10}\cr
4. & On the supremum of a nice class of partial sums.
\dotfill & \rightline{15}\cr
5. & Vapnik--\v{C}ervonenkis classes and $L_2$-dense classes of
functions. \dotfill & \rightline{25}\cr
6. & The proof of Theorems 4.1 and 4.2 on the supremum of random sums.
\dotfill & \vskip5pt \rightline{30} \cr
7. & The completion of the proof of Theorem 4.1.
\dotfill & \rightline{39}\cr
8. & Formulation of the main results of this work.
\dotfill & \rightline{48}\cr
9. & Some results about $U$-statistics.
\dotfill & \rightline{60}\cr
10. &  Multiple Wiener--It\^o integrals and their properties.
\dotfill & \rightline{74}\cr
11. & The diagram formula for products of degenerate $U$-statistics.
\dotfill & \rightline{90}\cr
12. & The proof of the diagram formula for $U$-statistics. \dotfill &
\rightline{102}\cr
13. & The proof of Theorems 8.3, 8.5 and Example 8.7. \dotfill &
\rightline{109}\cr
14. &  Reduction of the main result in this work. \dotfill &
\rightline{124}\cr
15. &  The strategy of the proof for the main result of this work.
\dotfill & \rightline{133}\cr
16. & A symmetrization argument. \dotfill & \rightline{141}\cr
17. & The proof of the main result. \dotfill & \rightline{156}\cr
18. &  An overview of the results in this work.
 \dotfill & \rightline{170}\cr
\noalign{\medskip}
&Appendix A.
The proof of some results about Vapnik--\v{C}ervonenkis classes.
\dotfill & \vskip5pt \rightline{184}\cr
& Appendix B.
The proof of the diagram formula for Wiener--It\^o integrals.
\dotfill &\vskip5pt \rightline{186}\cr
&Appendix C.
The proof of some results about Wiener--It\^o integrals.
\dotfill &\vskip5pt \rightline{194}\cr
& Appendix D.
The proof of Theorem 14.3. ( A result about the comparison of
$U$-statistics and decoupled $U$-statistics.)
\dotfill &\vskip5pt \rightline{202}\cr
\noalign{\medskip}
&References. \dotfill & \rightline{212}\cr
&Content. \dotfill & \rightline{215}\cr}}
$$
