text:
\begin{document}

\maketitle

\vspace{-0.67cm}

\begin{abstract}
We obtain Stein approximation bounds for stochastic integrals with respect to a Poisson random measure over $\real^d$, $d\geq 2$. This approach relies on third cumulant Edgeworth-type expansions based on derivation operators defined by the Malliavin calculus for Poisson random measures. The use of third cumulants can exhibit faster convergence rates than the standard Berry-Esseen rate for some sequences of Poisson stochastic integrals.
\end{abstract}
\noindent {\bf Key words:}
Stein approximation; multidimensional Poisson random measures; Poisson stochastic integrals; cumulants; Malliavin calculus; Edgeworth expansions.
\\
{\em Mathematics Subject Classification:} 62E17; 60H07; 60H05.

\baselineskip0.7cm

\section{Introduction}
Stein approximation bounds for stochastic integrals
with respect to a Poisson random measure have been obtained in
\cite{utzet2} using finite difference operators on the
Poisson space.
In this paper we derive related bounds for compensated Poisson stochastic integrals
$\delta ( u ) := \int_{\real^d} u_x ( \gamma ( dx ) - \lambda (dx))$
of compactly supported processes $(u_x)_{x\in \real^d}$ with respect to a Poisson
random measure $\gamma (dx)$ with intensity
the Lebesgue measure $\lambda (dx)$ on $\real^d$, $d\geq 2$.
In contrast with \cite{utzet2},
our approach is based on derivation operators
and Edgeworth type expansions that involve the
third cumulant of Poisson stochastic integrals,
and can result into faster convergence rates,
see e.g. \eqref{dlf} below.
\\

 Edgeworth type expansions have been obtained on the Wiener space in
 \cite{nourdinpeccati2009},
 \cite{campese}, by a construction of cumulant operators
 based on the inverse $L^{-1}$ of the
 Ornstein-Uhlenbeck operator,
 extending the results of \cite{nourdinpeccati}
 on Stein approximation and Berry-Esseen bounds.
\\

In Proposition~\ref{djklsad1} we derive Edgeworth type expansions of the form
\begin{equation}
\label{otf}
 E\left[
 \delta ( u )
 {{g}} ( \delta ( u ) )
 \right]
 =
 E\big[
 \Vert u \Vert_{L^2(\real^d)}^2 {{g}}' ( \delta ( u ) )
 \big]
 +
 \sum_{k=2}^n
 E\left[
 {{g}}^{(k)} ( \delta ( u ) )
 \Gamma_{k+1}^u {\bf 1}
 \right]
 +
 E\left[
 {{g}}^{(n+1)} ( \delta ( u ) )
 R_n^u
 \right]
\end{equation}
 when the random field $(u_x)_{x\in \real^d}$ is predictable with respect to
 a given total order on $\real^d$, where $\Gamma_k^u$ is a cumulant
 type operator and $R_n^u$ is a remainder term,
 defined using the derivation operators
 of the Malliavin calculus on the Poisson space.
\\

 Based on \eqref{otf}, in Corollary~\ref{djkd} we deduce Stein
 approximation bounds of the form
\begin{eqnarray}
\nonumber
 d ( \delta ( u ) , {\cal N} )
&  \leq &
 | 1 - \Var [ \delta (u) ] |
 +
 \sqrt{
 \Var \big[
 \Vert u \Vert_{L^2(\real^d )}^2
 \big]
 }
 \\
 \nonumber
 & &
 +
   E\left[
   \left|
   \int_{\real^d } u_x^3 \ \! \lambda ( dx )
 +
\left<
 u ,
 D
 \int_{\real^d } u^2_x \ \! \lambda ( dx )
 \right>_{L^2 (\real^d )}
 \right|
 \right]
+
   E\left[
 |
 R_1^u
 |
 \right]
,
\end{eqnarray}
 where $D$ is a gradient operator acting on Poisson functionals,
 and ${\cal N}\simeq {\cal N}(0,1)$ is a standard
 Gaussian random variable,
 see also Proposition~\ref{djhlkdsa}.
 Here,
$$
 d(F,G) :=
 \sup_{h\in {\cal L} }|\mathrm{E}[h( F )]-\mathrm{E}[h( G )]|
$$
 is the Wasserstein distance between
 the laws of two random variables $F$ and $G$,
 where ${\cal L}$ denotes the class of $1$-Lipschitz functions
 on $\real$.
\\

 In particular, when $f$ is a differentiable deterministic function with
 support in the closed centered ball $B(R):=B(0;R)$ with radius $R>0$
 we obtain bounds of the form
\begin{eqnarray}
\label{jkdf}
 d \left( \int_{\real^d} f(x) ( \gamma ( dx ) - \lambda (dx) ) , {\cal N} \right)
 & \leq &
 \big|
 1
 -
 \Vert
 f
 \Vert_{L^2(\real^d )}^2
 \big|
 +
 \left|
 \int_{\real^d} f^3(x) \ \! \lambda ( dx )
 \right|
 \\
 \nonumber
 & &
 +
 8 ( K_d v_d R )^2
  \Vert f \Vert_{L^2 (\real^d )}
  \Vert \nabla^{\real^d} f \Vert_{L^\infty (\real^d;\real^d)}^2,
\end{eqnarray}
where $v_d$ denotes the volume of the unit ball in $\real^d$
and $K_d>0$ is a constant depending only on $d\geq 2$.
 The bound \eqref{jkdf} can be compared to the classical Stein bound
\begin{equation}
 \label{e1}
 d \left( \int_{\real^d} f(x) ( \gamma ( dx ) - \lambda (dx) ) , {\cal N} \right)
 \leq
 \big|
 1
 -
 \Vert
 f
 \Vert_{L^2(\real^d )}^2
 \big|
 +
 \int_{\real^d} | f^3(x)| \ \! \lambda ( dx ),
 \end{equation}
for compensated Poisson stochastic integrals,
see Corollary~3.4 of \cite{utzet2},
which involves the $L^3(\real^d)$ norm of $f$ instead
of third cumulant $\kappa_3^f = \int_{\real^d} f^3(x) \ \! \lambda ( dx ) $
of $\int_{\real^d} f(x) ( \gamma ( dx ) - \lambda (dx) )$,
and relies on the use of finite difference operators,
see Theorem~3.1 of \cite{utzet2} and \S~4.2 of
\cite{bourguin}.
\\

For example when $f_k$, $k\geq 1$,
is a radial function given on $B(Rk^{1/d})$ by
$$
f_k (x) := \frac{1}{C \sqrt{k} } g\left(\frac{|x|_{\real^d}}{k^{1/d}}\right)
,
$$
where $g\in {\cal C}^1_0 ([0,R])$ is continuously differentiable on $[0,R]$,
and
$$
C^2 := \int_0^R g^2(r) r^{d-1} dr < \infty,
$$
so that $\Vert f_k \Vert_{L^2(B(Rk^{1/d}))}=1$,
 the bound \eqref{e1} yields the standard Berry-Esseen
 convergence rate
$$
 d \left( \int_{B(Rk^{1/d})} f_k(x) ( \gamma ( dx ) - \lambda (dx) ) , {\cal N} \right)
 \leq
\frac{v_d}{C^3 \sqrt{k}} \int_0^R |g (r)|^3 r^{d-1} dr,
\qquad k \geq 1.
$$
 While \eqref{jkdf} does not improve on \eqref{e1}
 when the function $f$ has constant sign, if
 $g$ satisfies the condition
$$
  \int_0^R g^3 (r) r^{d-1} dr =0,
$$
then the third cumulant bound
\eqref{jkdf} yields the $O(1/k)$ convergence rate
\begin{equation}
  \label{dlf}
 d \left( \int_{B(Rk^{1/d})} f_k(x) ( \gamma ( dx ) - \lambda (dx) ), {\cal N} \right)
 \leq
 \frac{ 2 ( 2 K_d v_d R )^2 d }{ k C^2}\Vert g' \Vert_\infty^2,
\end{equation}
which improves on the standard Berry-Esseen rate, see Section~\ref{s4}.
\\

In Sections~\ref{s2} and \ref{s2-1} we recall some background material
on the Malliavin calculus and differential geometry
on the Poisson space, by revisiting the approach
of \cite{pratprivault}, \cite{prconnection} using the recent
constructions of \cite{acosta} and references therein
on the solution of the divergence problem.
In Section~\ref{s3} we derive Edgeworth type expansions
for the compensated Poisson stochastic integral
$\delta (u)$,
based on a family of cumulant operators that are associated to the
random field $(u_x)_{x\in \real^d}$.
 In Section~\ref{s4} we derive Stein type approximation bounds for stochastic integrals,
  with deterministic examples.
\\

While this paper is dealing with Poisson random measures
on $\real^d$ with $d\geq 2$,
the special case $d=1$ requires a different treatment
for the standard Poisson process on the real half line,
see \cite{privaultthird}, and the $d$-dimensional setting of the present paper
shows significant differences with the one-dimensional case.
\subsection*{Preliminaries}
Let $d\geq 2$ and $0<R<R':=2R$.
We recall the existence of a ${\cal C}^\infty$ kernel function
$\G_\eta :B(R')\times B(R')\to \real^d$
defined as
$$
\G_\eta (x,y) : = \int_0^1 \frac{(x-y)}{s} \eta \left( y + \frac{x-y}{s} \right)
\frac{ds}{s^d},
\qquad
x,y \in B(R'),
$$
where $\eta\in {\cal C}^\infty_0 (B(R'))$ is such that
$\int_{B(R)} \eta(x) dx = 1$, see \cite{acosta}, and satisfying the following properties:
\begin{enumerate}[i)]
\item The kernel $\G_\eta (x,y)$ satisfies the bound
\begin{equation}
\label{gxy1}
|\G_\eta (x,y)|_{\real^d} \leq \frac{K_d}{|x-y|^{d-1}_{\real^d}},
\qquad
x,y \in B(R'),
\end{equation}
for a constant $K_d>0$ depending only on $d$,
see Lemma~2.1 of \cite{acosta},
by choosing $K_d$ and the function $\eta \in {\cal C}^\infty_c(B(R'))$ therein
so that $\Vert \eta\Vert_\infty \leq (d-1) K_d (R')^{-d}$.
\item For any $p>1$ and
  $g\in L^p(B(R'))$
  the function
\begin{equation}
\nonumber
f(x) : = \int_{B(R')} \G_\eta (x,y) g(y) \ \! \lambda (dy), \qquad x\in B(R'),
\end{equation}
 satisfies the bound
\begin{equation}
\label{gxy3}
\Vert f \Vert_{L^p(B(R');\real^d)}
\leq K_d v_d R' \Vert g \Vert_{L^p(B(R'))}, \qquad p>1,
\end{equation}
which follows from Young's inequality and \eqref{gxy1},
cf. Theorem~2.4 in \cite{acosta}.
\item For any $h\in {\cal C}^\infty_0 (B(R'))$
  we have the relation
\begin{equation}
\label{gxy4}
h(y) - \int_{B(R')\setminus B(R)} h(x) \eta(x) \ \! \lambda (dx) =  \int_{B(R')} \langle \G_\eta (x,y) , \nabla^{\real^d}_x h(x) \rangle_{\real^d} \lambda (dx),
\quad
y\in B(R'),
\end{equation}
cf. Lemma~2.2 in \cite{acosta}, by taking
$\eta\in {\cal C}^\infty_c(B(R')\setminus B(R))$.
In particular, when $h\in {\cal C}^\infty_0 (B(R))$ we have
\begin{equation}
\label{gxy4-2}
h(y) =  \int_{B(R')} \langle \G_\eta (x,y) , \nabla^{\real^d}_x h(x) \rangle_{\real^d}
\lambda (dx), \qquad y\in B(R').
\end{equation}
\end{enumerate}
\noindent
 An extension of the framework of this paper
 by replacing $B(R)$ with a compact $d$-dimensional Riemannian manifold $M$
 and $\lambda (dx)$ with the volume element of $M$
 requires the Laplacian ${\cal L} = \div^M\nabla^M$ to be
 invertible on ${\cal C}_c^\infty (M)$, with
$${\cal L}^{-1} u (x) = \int_M \g (x,y) u(y)\ \! \lambda (dy),
 \qquad x\in M, \ u\in {\cal C}^\infty_c (M),
$$
 where $\g (x,y)$ is the heat kernel on $M$.
 In this case we can define $\G_\eta (x,y) \in \real^d$ as
$$
 \G_\eta (x,y) = \nabla^M_x \g (x,y), \qquad {\lambda} \otimes {\lambda} (dx,dy)-a.e.
$$
 with the relation
$$
 \nabla^M_x {\cal L}^{-1} u (x)
 = \int_M u (y) \G_\eta (x,y) \ \! \lambda (dy)\in T_xM, \qquad x\in M, \ u \in {\cal C}^\infty_c (M),
$$
 from which the divergence inversion relation \eqref{gxy4-2} holds
 by duality.
\section{Gradient, divergence and covariance derivative}
\label{s2}
There exists different notions of gradient and divergence
operators for functionals of Poisson random measures.
The operators of \cite{akr}, \cite{privaulttorrisi}, \cite{clausel},
and their associated integration
by parts formula rely on an $\real^d$-valued gradient
for random functionals and a divergence operator which is
associated to the non-compensated Poisson stochastic
integral of the divergence of $\real^d$-valued random fields.
This particularity, together with a lack of a suitable commutation relation
between gradient and divergence operators on Poisson functionals,
makes this framework difficult to use for a direct analysis of Poisson stochastic
integrals, while it has found applications to statistical estimation
and sensitivity analysis \cite{clausel}, \cite{privaulttorrisi}.
\\

In this paper we use the construction of \cite{pratprivault},
\cite{prconnection} which relies on real-valued tangent processes
and of a divergence operator that directly
extends the compensated Poisson stochastic integral.
This framework also allows for simple commutation relations
between gradient and divergence operators using
the deterministic inner product in $L^2(\real^d,\lambda)$,
see Proposition~\ref{commut}, and it naturally involves
the Poisson cumulants, see Definition~\ref{def01} and Relation~\eqref{ifinad2}.
\subsubsection*{Gradient operator}
In the sequel we consider a Poisson random measure
$\gamma (dx )$ on a probability space
$(\Omega , {\cal F} , P)$ and we
let $\{X_1,\ldots ,X_n\}$ denote the configuration points
 of $\gamma (dx )$ when $B(R)$ contains $n$ points in the configuration $\gamma$, i.e.
 when $\gamma (B(R))=n$.
\begin{definition}
 Given $A$ a closed subset of $B(R')$, we let ${\cal S}_A$ denote the
set of random functionals $F_A$ of the form
\begin{equation}
  \label{f}
 F_A =
  \sum_{n=0}^\infty
  {\bf 1}_{\{ \gamma ( B(R))=n\} }
 f_n \left( X_1,\ldots , X_n \right)
 ,
\end{equation}
where
 $f_0\in \real$ and $(f_n)_{n\geq 1}$ is a sequence
of functions
satisfying the following conditions:
\begin{enumerate}[-]
\item
  for all $n \geq 1$, $f_n \in {\cal C}^\infty_c ( A^n )$
  is a symmetric function in $n$ variables,
\item
 for all $n\geq 1$ and $i=1,\ldots , n$ we have the continuity condition
\begin{equation}
   \label{cont}
 f_n \left( x_1,\ldots , x_n \right)
 =
 f_{n-1} \left( x_1,\ldots , x_{i-1},x_{i+1}, \ldots , x_n \right)
 ,
\end{equation}
 for all $x_1,\ldots , x_n\in B(R')$ such that $|x_i|_{\real^d} \geq R$.
\end{enumerate}
 We also let ${\cal S}$ denote the union of the
 sets ${\cal S}_A$ over the closed subsets $A$ of $B(R')$.
\end{definition}
The gradient operator $D$ is defined on random functionals
 $F \in {\cal S}$ of the form \eqref{f} as
\begin{equation}
\label{ndfgd}
D_y F :=
  \sum_{n=1}^\infty
   {\bf 1}_{\{ \gamma ( B(R))=n\} }
 \sum_{i=1}^n
 \langle \G_\eta (X_i,y) ,
 \nabla^{\real^d}_{x_i} f\left( X_1,\ldots , X_n \right)
 \rangle_{\real^d}
,
\end{equation}
$y\in B(R)$.
For any $F\in {\cal S}$, by \eqref{gxy1}
we have $DF\in L^1(\Omega \times B(R))$ from the bound
\begin{eqnarray*}
  E \left[ \int_{B(R)} | D_x F | \ \! \lambda (dx) \right]
  & \leq &
\Vert | \nabla^{\real^d} f|_{\real^d} \Vert_\infty
 E \left[
   \int_{B(R)} \int_{B(R)} | \G_\eta (x,y) |_{\real^d}
  \gamma (dx) \lambda (dy)
 \right]
 \\
 & = &
\Vert | \nabla^{\real^d} f|_{\real^d} \Vert_\infty
 \int_{B(R)} \int_{B(R)} | \G_\eta (x,y) |_{\real^d}
   \lambda (dx)
   \lambda (dy)
\\
 & = &
    K_d  \Vert | \nabla^{\real^d} f|_{\real^d} \Vert_\infty
 \int_{B(R)} \int_{B(R)} \frac{1}{|x-y|_{\real^d}^{d-1}}
    \lambda ( dx ) \lambda (dy)
  \\
 & \leq &
 K_d v_d^2 R' R^d
\Vert | \nabla^{\real^d} f|_{\real^d} \Vert_\infty
 \\
 & < & \infty.
\end{eqnarray*}
\subsubsection*{Poisson-Skorohod integral}
 We let ${\cal U}_0$ denote the space of simple random fields of the form
\begin{equation}
\label{u}
 u = \sum_{i=1}^{n} g_i G_i, \qquad n\geq 1,
\end{equation}
with $G_i \in {\cal S}_{A_i}$
and $g_i \in {\cal C}^\infty_0 (B(R))$,
 $i=1,\ldots ,n$.
  \begin{definition}
 We define the Poisson-Skorohod integral $\delta (u)$ of $u \in {\cal U}_0$
 of the form \eqref{u} as
\begin{equation}
\label{dltu}
 \delta(u) := \sum_{i=1}^n
 \Big( G_i \int_{B(R)} g_i (x) ( \gamma (dx) - \lambda (dx) )
 - \langle g_i , DG_i \rangle_{L^2(B(R))} \Big).
\end{equation}
\end{definition}
 In particular, for $h\in {\cal C}^\infty_0 (B(R))$
  we have
$$
 \delta (h) = \int_{B(R)} h(x) ( \gamma (dx) - \lambda (dx) ).
$$
 The proof of the next proposition, cf.
 Proposition~8.5.1 in \cite{pratprivault}
 and Proposition~5.1 in \cite{prconnection},
 is given in the appendix.
\begin{prop}
  \label{prop1}
  The operators $D$ and $\delta$ satisfy the duality relation
\begin{equation}
  \label{jldf}
  E[ \langle u , DF\rangle_{L^2(B(R))} ]
 =
 E [ F \delta ( u ) ],
\qquad F\in {\cal S},
 \qquad u\in {\cal U}_0.
\end{equation}
\end{prop}
As a consequence of Proposition~\ref{prop1}
and the denseness of ${\cal S}$ in $L^1(\Omega )$
and that of ${\cal U}_0$ in $L^1 (\Omega \times B(R) )$,
the gradient operator $D$ is
 closable in the sense that if $(F_n)_{n\in\inte}\subset {\cal S}$
 tends to zero in $L^2 (\Omega )$
 and $(DF_n)_{n\in\inte}$ converges
 to $U$ in $L^1(\Omega \times B(R))$, then $U=0$ a.e..
 Similarly, the divergence operator $\delta$ is
 closable in the sense that if $(u_n)_{n\in\inte}\subset {\cal U}_0$
 tends to zero in $L^2 (\Omega \times B(R))$
 and $(\delta ( u_n ))_{n\in\inte}$ converges
 to $G$ in $L^1 (\Omega )$, then $G=0$ a.s..
\\

 The gradient operator $D$ defines the Sobolev space
 $\dee^{1,1}$ with the Sobolev norm
$$
 \Vert F \Vert_{\dee_{1,1} }
 :=
 \Vert F \Vert_{L^2 ( \Omega )}
 +
 \Vert D F \Vert_{L^1 ( \Omega  \times B(R) )}
,
 \qquad
 F \in {\cal S}.
$$
 In the sequel we fix a total order $\preceq$ on $B(R)$ and
 consider the space ${\cal P}_0\subset {\cal U}_0$
  of simple predictable random field of the form
\begin{equation}
\label{uf}
u: = \sum_{i=1}^n g_iF_i,
\end{equation}
such that the supports of $g_1,\ldots , g_n$ satisfy
$$
\Supp ( g_i ) \preceq \cdots \preceq \Supp ( g_n )
\quad
\mbox{and}
\quad
F_i \in {\cal S}_{A_i},
$$
where $\Supp ( g_1 )\cup \cdots \cup \Supp ( g_{i-1} ) \subset A_i\subset B(R')$
and $A_i \preceq \Supp ( g_i )$, $i=1,\ldots , n$.
\\

Such random fields are predictable in the sense of e.g. \S~5 of
\cite{last} and references therein.
\\

We will also assume that the order $\preceq$ is compatible with the
kernel $\G_\eta $ in the sense that
\begin{equation}
  \label{comp}
  \G_\eta (x,y) = 0
  \quad
  \mbox{for all} \quad
  x, y \in B(R)
  \quad
  \mbox{such that} \quad
  x \preceq y.
\end{equation}
 Under the compatibility condition \eqref{comp} we have in particular
$$
D_y F = 0, \quad
y\in B(R),
\quad
A\preceq y,
\qquad
F\in {\cal S}_A.
$$
Moreover, if $u \in {\cal P}_0$ is a predictable random field
of the form \eqref{uf} we note that by \eqref{ndfgd} and the compatibility condition
\eqref{comp} we have
$$
D_yF_i=0, \qquad A_i \preceq y, \quad i=1,\ldots , n,
$$
 hence
\begin{equation}
\label{comp2}
D_y u_x = 0 ,
\qquad
x \preceq y,
\quad
x,y \in B(R).
\end{equation}
{\em Example}.
The order $\preceq$ defined by
\begin{equation}
  \label{compat}
x = (x^{(1)},\ldots , x^{(d)}) \preceq y = (y^{(1)},\ldots , y^{(d)})
\quad
\Longleftrightarrow
\quad
x^{(1)} \leq y^{(1)}
\end{equation}
is compatible with the
kernel $\G_\eta$ provided that the support of
$\eta$ is contained in
$$
\big\{
x = (x^{(1)},\ldots , x^{(d)}) \in B(R')\setminus B(R) \
: \
x^{(1)}>R \big\}.
$$

The proof of the next Proposition~\ref{prop2}
is given in the appendix.
\begin{prop}
\label{prop2}
 The Poisson-Skorohod integral of
 $u=(u_x)_{x\in B( R )}$ in the space ${\cal P}_0$ of simple predictable
 random fields satisfies the relation
\begin{equation}
  \label{rel}
\delta ( u ) = \int_{B(R)} u_x ( \gamma ( dx ) - \lambda ( dx ) ),
\end{equation}
which extends to the closure of ${\cal P}_0$ in $L^2(\Omega \times B(R))$
by density and the isometry relation
\begin{equation}
  \label{isomrel}
E [ \delta (u)^2 ]
=
E\left[
\int_{B(R)} u^2_x \ \! \lambda ( dx )
  \right], \qquad u \in {\cal P}_0.
\end{equation}
\end{prop}
\subsubsection*{Covariant derivative}
 In addition to the gradient operator $D$,
 we will also need the following notion of covariant
 derivative operator $\widetilde{\nabla}$ defined on
 stochastic processes that are viewed as tangent processes on the Poisson space
 $\Omega$, see \cite{prconnection}.
\begin{definition}
  Let the operator $\widetilde{\nabla}$ be defined
  on $u \in {\cal P}_0$ as
\begin{equation}
\nonumber
 \widetilde{\nabla}_y u_x : =
 {D}_y u_x + \langle \G_\eta (x,y) , \nabla^{\real^d}_x u_x \rangle_{\real^d},
 \qquad x, y \in B(R).
\end{equation}
\end{definition}
We note that from the compatibility condition \eqref{comp}
and Relation~\eqref{comp2} we also have
\begin{equation}
\label{comp3}
\widetilde{\nabla}_y u_x = 0 ,
\qquad
x \preceq y,
\quad
x,y \in B(R).
\end{equation}
From the bound
\begin{eqnarray*}
  \lefteqn{
    \!  \!  \!  \!  \!  \!  \!  \!  \!  \!  \!
    E\left[ \int_{B(R)\times B(R)}| \widetilde{\nabla}_x u_y | \ \! \lambda ( dx ) \lambda (dy) \right]
  }
  \\
  & \leq &
\Vert D u \Vert_{L^1(\Omega \times B(R) \times B(R) )}
  +
  E\left[
    \int_{B(R) \times B(R) } | \langle \G_\eta (x,y) , \nabla^{\real^d}_x u_x \rangle_{\real^d} |
    \ \! \lambda ( dx ) \lambda (dy)
    \right]
  \\
  & \leq &
 \Vert D u \Vert_{L^1(\Omega \times B(R) \times B(R) )}
  +
  K_d E\left[
    \int_{B(R) \times B(R) } \frac{1}{|x-y|_{\real^d}^{d-1}}
    | \nabla^{\real^d} u_x |_{\real^d}
    \lambda ( dx ) \lambda (dy)
    \right]
  \\
  & \leq &
 \Vert D u \Vert_{L^1(\Omega \times B(R) \times B(R) )}
  + K_d v_d R'
  E\left[
    \int_{B(R)}
    | \nabla^{\real^d}_x u_x |_{\real^d}
    \lambda ( dx )
    \right]
  \\
  & = &
 \Vert D u \Vert_{L^1(\Omega \times B(R) \times B(R) )}
  + K_d v_d R' \Vert \nabla^{\real^d} u \Vert_{L^1(\Omega \times B(R) ; \real^d)},
\end{eqnarray*}
we check that $\widetilde{\nabla}$ extends to the Sobolev space
$\widetilde{\dee}^{1,1}_0$ of {\em predictable} random fields defined
as the completion of
${\cal P}_0$ under the Sobolev norm
$$
 \Vert u \Vert_{ \widetilde{\dee}^{1,1}}
 : =
 \Vert u \Vert_{L^2(\Omega , W^{1,1}_0(B(R)))}
 + \Vert D u \Vert_{L^1(\Omega \times B(R)\times B(R) ) }
 ,
 \qquad u \in {\cal P}_0,
$$
 where $W^{1,p}_0 (B(R))$ is the first order Sobolev space completion
 of ${\cal C}^\infty_0(B(R))$ under the norm
$$
 \Vert f \Vert_{W^{1,p}(B(R))}
 : =
 \Vert f \Vert_{L^p(B(R))}
 +
 \Vert \nabla^{\real^d} f \Vert_{L^p(B(R);\real^d)}, \qquad p \geq 1.
$$
\subsubsection*{Commutation relation}
 In the sequel, we denote by $\widetilde{\dee}^{1,\infty}_0$ the set of
 predictable random fields $u$ in $\widetilde{\dee}^{1,1}_0$
 that are bounded together with their covariant derivative
 $\widetilde{\nabla}u$.
\begin{prop}
\label{commut}
For $u \in \widetilde{\dee}^{1,\infty}_0$ a predictable random field,
we have the commutation relation
\begin{equation}
\label{com}
D_y \delta(u) = u(y) + \delta(\widetilde{\nabla}_y u),
\qquad y\in B(R).
\end{equation}
\end{prop}
\begin{Proof}
 Taking $h\in {\cal C}^\infty_0 (B(R))$, we have $\delta( h) \in {\cal S}$ and
\begin{eqnarray*}
 D_y \delta( h) & = &
 D_y \int_{B(R)} h(y) ( \gamma (dx ) - \lambda ( dx ) )
\\
 & = &
 \int_{B(R)} \langle \G_\eta (x,y) , \nabla^{\real^d}_x h(x) \rangle_{\real^d} \gamma (dx)
\\
 & = &
  \int_{B(R)} \langle \G_\eta (x,y) , \nabla^{\real^d}_x h(x) \rangle_{\real^d} \lambda (dx)
 +
 \delta ( \widetilde{\nabla}_y h )
 \\
 & = &
 h(y) + \delta ( \widetilde{\nabla}_y h ).
\end{eqnarray*}
 where we applied \eqref{gxy4-2}.
 Next, taking $u=hF \in {\cal P}_0$ a simple predictable random field,
 we check that $\delta( u) \in {\cal S}$, and by \eqref{dltu} or \eqref{dltu2} we have
\begin{eqnarray*}
 D_y \delta( F h) & = &
 D_y \left( F \delta( h) - \langle h , DF\rangle_{L^2(B(R))} \right)
 \\
 & = &
 D_y \left( F \delta( h) \right)
 \\
 & = &
 \delta( h) D_y F + F D_y \delta( h)
 \\
 & = &
 \delta( h) D_y F + F ( h(y) + \delta ( \widetilde{\nabla}_y h ) )
 \\
 & = &
 F h(y) + \delta( h D_y F + F \widetilde{\nabla}_y h )
 \\
 & = &
 F h(y)
 +
  \delta( \widetilde{\nabla}_y ( F h ) )
\\
 & = &
 u_y + \delta( \widetilde{\nabla}_y u ) ,
 \qquad y \in B(R).
\end{eqnarray*}
We conclude by the denseness of ${\cal P}_0$ in $\widetilde{\dee}^{1,1}_0$
and by the closability of $\widetilde{\nabla}$, $D$ and $\delta$.
\end{Proof}
\section{Cumulant operators}
\label{s2-1}
In the sequel, given $h$ in the standard Sobolev space
$W^{1,p}( B(R) )$ on $B(R)$ and $f \in L^q(B(R))$
with $1=p^{-1}+q^{-1}$, $p,q\in [1,\infty]$, we define
\begin{equation}
\label{1lpa}
 ( \widetilde{\nabla} h ) f_x
: =
 \int_{B(R)} f(y) \widetilde{\nabla}_y h(x) \ \! \lambda (dy)
 =
 \int_{B(R)} f(y) \langle \G_\eta (x,y) , \nabla^{\real^d}_x h(x) \rangle_{\real^d} \lambda (dy)
,
\end{equation}
$x \in B(R)$.
More generally, given $k\geq 1$ and $u \in \widetilde{\dee}^{1,1}_0$
a predictable random field, we let the operator $( \widetilde{\nabla} u)^k$
be defined in the sense of matrix powers with continuous indices,
as
\begin{equation}
\nonumber
 ( \widetilde{\nabla} u )^k
 f_y
 =
 \int_{B(R)}
 \cdots
 \int_{B(R)}
 (
 \widetilde{\nabla}_{x_k} u_y
 \widetilde{\nabla}_{x_{k-1}} u_{x_k}
 \cdots
 \widetilde{\nabla}_{x_1} u_{x_2}
 )
 f_{x_1}
 \ \! \lambda ( dx_1 ) \cdots \lambda ( dx_k),
\end{equation}
$y \in B(R)$, $f \in L^2(B(R))$.
\begin{prop}
  \label{p1}
  For any $n \in \inte$, $p > 1$, $r\in [0,1]$,
  $h\in W^{1,p/(1-r)^{n-1}/r}( B(R) )$ and $f \in L^{p/(1-r)^n} (B(R))$ we have the bound
\begin{equation}
\label{vrt}
  \Vert ( \widetilde{\nabla} h )^n f \Vert_{L^p(B(R))}
 \leq
  ( K_d v_d R' )^n
  \Vert f \Vert_{L^{p/(1-r)^n}(B(R))}
  \prod_{j=1}^n \Vert \nabla^{\real^d} h \Vert_{L^{p/(1-r)^{j-1}/r}(B(R);\real^d)}.
\end{equation}
\end{prop}
\begin{Proof}
 For $n=1$ we have
\begin{eqnarray}
  \nonumber
  \lefteqn{
    \Vert ( \widetilde{\nabla} h ) f \Vert_{L^p(B(R))}^p
  =
  \int_{B(R)} \left| \int_{B(R)} f(y) \widetilde{\nabla}_y h(x) \ \! \lambda (dy) \right|^p
  \lambda (dx)
  }
  \\
\nonumber
    & = &
  \int_{B(R)} \left| \int_{B(R)} f(y)
  \langle \G_\eta (x,y) , \nabla^{\real^d}_x h(x) \rangle_{\real^d}
  \lambda (dy) \right|^p
  \lambda (dx)
  \\
\nonumber
    & = &
  \int_{B(R)} \left| \left<
  \int_{B(R)} f(y)
  \G_\eta (x,y) \ \! \lambda (dy)
  , \nabla^{\real^d}_x h(x) \right>_{\real^d}
  \right|^p
  \lambda (dx)
  \\
\nonumber
    & \leq &
  \int_{B(R)}
  \left|
  \int_{B(R)} f(y)
  \G_\eta (x,y) \ \! \lambda (dy)
  \right|_{\real^d}^p
  | \nabla^{\real^d}_x h(x) |_{\real^d}^p
  \lambda (dx)
  \\
\nonumber
    & = &
  \left(
    \int_{B(R)}
  \left|
  \int_{B(R)} f(y)
  \G_\eta (x,y) \ \! \lambda (dy)
  \right|_{\real^d}^{p/(1-r)}
  \lambda (dx)
  \right)^{1-r}
  \left(
    \int_{B(R)}
  | \nabla^{\real^d}_x h(x) |_{\real^d}^{p/r}
  \lambda (dx)
 \right)^r
  \\
  \label{djcdfd}
  & \leq &
  ( K_d v_d R' )^p
  \Vert f \Vert_{L^{p/(1-r)}(B(R))}^p
  \Vert \nabla^{\real^d} h \Vert_{L^{p/r}(B(R);\real^d)}^p,
  \end{eqnarray}
 where we used the bound \eqref{gxy3}.
 Next, assuming that \eqref{vrt} holds at the rank $n\geq 1$
 and using \eqref{djcdfd}, we have
\begin{eqnarray*}
  \Vert ( \widetilde{\nabla} h )^{n+1} f \Vert_{L^p(B(R))}
  & = &
  \Vert ( \widetilde{\nabla} h )^n ( \widetilde{\nabla} h ) f \Vert_{L^p(B(R))}
  \\
   & \leq &
  ( K_d v_d R' )^n
  \Vert ( \widetilde{\nabla} h ) f \Vert_{L^{p/(1-r)^n}(B(R))}
  \prod_{j=1}^n \Vert \nabla^{\real^d} h \Vert_{L^{p/(1-r)^{j-1}/r}(B(R);\real^d)}
    \\
   & \leq &
  ( K_d v_d R' )^{n+1}
  \Vert f \Vert_{L^{p/(1-r)^{n+1}} (B(R))}
  \prod_{j=1}^{n+1} \Vert \nabla^{\real^d} h \Vert_{L^{p/(1-r)^{j-1}/r}(B(R);\real^d)},
\end{eqnarray*}
 and we conclude to \eqref{vrt} by induction.
\end{Proof}
In particular, for $r=0$, $f\in L^p (B(R))$, $p>1$,
and $h\in W^{1,1}(B(R))$ the argument of Proposition~\ref{p1}
shows that
$$
 \Vert ( \widetilde{\nabla} h )^n f \Vert_{L^p(B(R))}
 \leq
  ( K_d v_d R' )^n
  \Vert f \Vert_{L^p (B(R))}
  \Vert \nabla^{\real^d} h \Vert_{L^\infty (B(R);\real^d)}^n,
  \qquad  n \in \inte.
$$
 We note that for $u \in \widetilde{\dee}^{1,\infty}_0$ a
 predictable random field, the
 random field $( \widetilde{\nabla} u) u \in \widetilde{\dee}^{1,\infty}_0$
 is also predictable from \eqref{comp3} and \eqref{1lpa}.
\\

 In the next definition we construct a family of cumulant operators
    which differs from the one introduced in \cite{nourdin10} on the Wiener space.
\begin{definition}
\label{def01}
Given $k\geq 2$ and $u \in \widetilde{\dee}^{1,\infty}_0$ a
predictable random field
we define the operators
$\Gamma^u_k : \dee_{1,1} \longrightarrow L^1( \Omega )$ by
$$
 \Gamma^u_k F
 : =
 F
 \langle
 ( \widetilde{\nabla} u )^{k-2} u
 ,
 u
 \rangle_{L^2(B(R))}
 +
 \langle
 ( \widetilde{\nabla} u )^{k-1} u
 ,
 DF
 \rangle_{L^2(B(R))}
 ,
 \quad
F\in \dee_{1,1}.
$$
\end{definition}
  We note that for
  $h$ in the space $W^{1,\infty}( B(R) )$ of bounded functions in
  $W^{1,1}( B(R) )$, and $f \in L^p (B(R))$, $p>1$,
  $m\geq 1$, we have
\begin{eqnarray*}
 \langle h^m , ( \widetilde{\nabla} h ) f \rangle_{L^2(B(R))}
 & = &
 \int_{B(R)} h^m(x)  \int_{B(R)} f(y) \langle \G_\eta (x,y)  , \nabla^{\real^d}_x h(x) \rangle_{\real^d}
 \lambda (dy)
 \lambda (dx)
 \\
  & = &
 \frac{1}{m+1}
 \int_{B(R)} \int_{B(R)} f(y) \langle \G_\eta (x,y)  , \nabla^{\real^d}_x h^{m+1} (x) \rangle_{\real^d}
 \lambda (dy)
 \lambda (dx)
 \\
  & = &
 \frac{1}{m+1}
 \int_{B(R)}
 f(x) h^{m+1} (x)
 \ \! \lambda (dx),
\end{eqnarray*}
where we applied \eqref{gxy4}, hence
$$
 \langle h^m , ( \widetilde{\nabla} h )^{n+1} f \rangle_{L^2(B(R))}
 =
 \frac{1}{m+1}
 \int_{B(R)}
 h^{m+1} (x) ( \widetilde{\nabla} h )^n f (x)
 \ \! \lambda (dx)
 ,
 $$
 which implies by induction
 $$
 \langle ( \widetilde{\nabla} h )^n f , h^m \rangle_{L^2(B(R))}
 =
 \frac{m!}{(m+n)!}
 \int_{B(R)}
 h^{m+n} (x) f (x)
 \ \! \lambda (dx).
$$
 In Lemma~\ref{itlw2.1} we generalize this identity to $h$ a random field.
\begin{lemma}
\label{itlw2.1}
For $n\in \inte$, $m \geq 1$, $u \in \widetilde{\dee}^{1,\infty}_0$
a predictable random field and $f \in L^p (B(R))$, $p>1$, we have
\begin{eqnarray}
\label{djklll}
 \langle
 ( \widetilde{\nabla} u )^n f
 ,
 u^m
 \rangle_{L^2(B(R))}
 & = &
 \frac{m!}{(m+n)!}
 \int_{B(R)}
 u_x^{m+n} f (x) \ \! \lambda ( dx )
\\
 \nonumber
 & &
 + \sum_{k=1}^n
 \frac{m!}{(m+k)!}
 \left<
 ( \widetilde{\nabla} u )^{n-k}
 f  ,
 D
 \int_{B(R)}
 u^{m+k}_x \ \! \lambda ( dx )
 \right>_{L^2(B(R))}.
\end{eqnarray}
\end{lemma}
\begin{Proof}
  Using the adjoint $\widetilde{\nabla}^* u$ of $\widetilde{\nabla} u$ on $L^2(B(R))$
  given by
$$
 ( \widetilde{\nabla}^* u) v_y : = \int_{B(R)} ( \widetilde{\nabla}_y  u_x ) v_x \ \! \lambda ( dx ),
 \qquad
 y \in B(R), \quad
 v \in L^2(B(R)),
$$
 with the duality relation
$$
 \langle v , ( \widetilde{\nabla}^* u ) h \rangle_{L^2(B(R))}
 =
  \langle ( \widetilde{\nabla} u ) v , h \rangle_{L^2(B(R))},
 \qquad
 h ,v \in L^2(B(R)),
$$
 we will show by induction on $k=0,1,\ldots, n$ that
\begin{align}
\nonumber
& ( \widetilde{\nabla}^* u )^n u^m_{x_0}
 =
 \int_{B(R)}
 \cdots
 \int_{B(R)}
 u^m_{x_n}
 \widetilde{\nabla}_{x_0} u_{x_1}
 \widetilde{\nabla}_{x_1} u_{x_2}
 \cdots
 \widetilde{\nabla}_{x_{n-1}} u_{x_n}
 \ \! \lambda ( dx_1 ) \cdots \lambda ( dx_n )
\\
 \nonumber
 & =
\sum_{i=1}^k
 \frac{m!}{(m+i)!}
 \int_{B(R)}
 \cdots
 \int_{B(R)}
 \widetilde{\nabla}_{x_0} u_{x_1}
 \cdots
 \widetilde{\nabla}_{x_{n-i-1}} u_{x_{n-i}}
 D_{x_{n-i}}
 u_{x_{n+1-i}}^{m+i}
 \ \! \lambda ( dx_1 ) \cdots \lambda ( dx_{n-i-1} )
\\
\label{id}
 &
 +
 \frac{m!}{(m+k)!}
 \int_{B(R)}
 \cdots
 \int_{B(R)}
 u_{x_{n-k}}^{m+k}
 \widetilde{\nabla}_{x_0} u_{x_1}
 \cdots
 \widetilde{\nabla}_{x_{n-k-1}} u_{x_{n-k}}
 \ \! \lambda ( dx_1 ) \cdots \lambda ( dx_{n-k} ).
\end{align}
 By \eqref{1lpa}, this relation holds for $k=0$.
 Next, assuming that the identity \eqref{id} holds for some
 $k \in \{ 0,1, \ldots ,n-1 \}$, and using the relation
$$
 \widetilde{\nabla}_{x_{n-k-1}} u_{x_{n-k}}
 =
 D_{x_{n-k-1}} u_{x_{n-k}}
 +
 \langle  \G_\eta (x_{n-k}, x_{n-k-1} ) , \widetilde{\nabla}_{x_{n-k}} u_{x_{n-k}} \rangle_{\real^d},
 \quad
 x_{n-k-1} , x_{n-k} \in B(R),
$$
 we have
\begin{align*}
& ( \widetilde{\nabla}^* u )^n u_{x_0}
  \\
  &
  =
 \sum_{i=1}^k
 \frac{m!}{(m+i)!}
 \int_{B(R)}
 \cdots
 \int_{B(R)}
 \widetilde{\nabla}_{x_0} u_{x_1}
 \cdots
 \widetilde{\nabla}_{x_{n-i-1}} u_{x_{n-i}}
 D_{x_{n-i}}
 u_{x_{n+1-i}}^{m+i}
 \ \! \lambda ( dx_1 ) \cdots \lambda ( x_{n+1-i} )
\\
 &
 +
 \frac{m!}{(m+k)!}
 \int_{B(R)}
 \cdots
 \int_{B(R)}
 u_{x_{n-k}}^{m+k}
 \widetilde{\nabla}_{x_0} u_{x_1}
 \cdots
 \widetilde{\nabla}_{x_{n-k-1}} u_{x_{n-k}}
 \ \! \lambda ( dx_1 ) \cdots \lambda ( dx_{n-k} )
\\
 & =
 \sum_{i=1}^k
 \frac{m!}{(m+i)!}
 \int_{B(R)}
 \cdots
 \int_{B(R)}
 \widetilde{\nabla}_{x_0} u_{x_1}
 \cdots
 \widetilde{\nabla}_{x_{n-i-1}} u_{x_{n-i}}
 D_{x_{n-i}}
 u_{x_{n+1-i}}^{m+i}
 \ \! \lambda ( dx_1 ) \cdots \lambda ( dx_{n+1-i} )
\\
 &
 +
 \frac{m!}{(m+k)!}
 \int_{B(R)}
 \cdots
 \int_{B(R)}
 u_{x_{n-k}}^{m+k}
 \widetilde{\nabla}_{x_0} u_{x_1}
 \cdots
 \widetilde{\nabla}_{x_{n-k-2}} u_{x_{n-k-1}}
 D_{x_{n-k-1}} u_{x_{n-k}}
 \ \! \lambda ( dx_1 ) \cdots \lambda ( dx_{n-k} )
\\
 &
 +
 \frac{m!}{(m+k)!}
 \int_{B(R)}
 \cdots
 \int_{B(R)}
 \langle  \G_\eta (x_{n-k} , x_{n-k-1} ) , \widetilde{\nabla}_{x_{n-k}} u_{x_{n-k}} \rangle_{\real^d}
 \\
 & \qquad
 \times
 u_{x_{n-k}}^{m+k-2}
 \widetilde{\nabla}_{x_0} u_{x_1}
 \cdots
 \widetilde{\nabla}_{x_{n-2-k}} u_{x_{n-k-1}}
 \ \! \lambda ( dx_1 ) \cdots \lambda ( dx_{n-k} )
\\
 & =
 \sum_{i=1}^k
 \frac{m!}{(m+i)!}
 \int_{B(R)}
 \cdots
 \int_{B(R)}
 \widetilde{\nabla}_{x_0} u_{x_1}
 \cdots
 \widetilde{\nabla}_{x_{n-i-1}} u_{x_{n-i}}
 D_{x_{n-i}}
 u_{x_{n+1-i}}^{m+i}
 \ \! \lambda ( dx_1 ) \cdots \lambda ( dx_{n+1-i} )
\\
 &
 +
 \frac{m!}{(m+k+1)!}
 \int_{B(R)}
 \cdots
 \int_{B(R)}
 \widetilde{\nabla}_{x_0} u_{x_1}
 \cdots
 \widetilde{\nabla}_{x_{n-k}} u_{x_{n-k-1}}
 D_{x_{n-k-1}} u_{x_{n-k}}^{m+k+1}
 \ \! \lambda ( dx_1 ) \cdots \lambda ( dx_{n-k} )
\\
 &
 +
 \frac{m!}{(m+k+1)!}
 \int_{B(R)}
 \cdots
 \int_{B(R)}
 \widetilde{\nabla}_{x_0} u_{x_1}
 \cdots
 \widetilde{\nabla}_{x_{n-k-2}} u_{x_{n-k-1}}
 \\
 & \qquad
 \times \int_{B(R)}
 \langle  \G_\eta (x , x_{n-k-1} ) , \nabla^{\real^d}_x u^{m+k+1}_x \rangle_{\real^d}
 \lambda ( dx )
 \lambda ( dx_1 ) \cdots \lambda ( dx_{n-k-1} )
\\
 & =
 \sum_{i=1}^{k+1}
 \frac{m!}{(m+i)!}
 \int_{B(R)}
 \cdots
 \int_{B(R)}
 \widetilde{\nabla}_{x_0} u_{x_1}
 \cdots
 \widetilde{\nabla}_{x_{n-i-1}} u_{x_{n-i}}
 D_{x_{n-i}}
 u_{x_{n+1-i}}^{m+i}
 \ \! \lambda ( dx_1 ) \cdots \lambda ( dx_{n+1-i} )
\\
 &
 +
 \frac{m!}{(m+k+1)!}
 \int_{B(R)}
 \cdots
 \int_{B(R)}
 u_{x_{n-k-1}}^{m+k+1}
 \widetilde{\nabla}_{x_0} u_{x_1}
 \cdots
 \widetilde{\nabla}_{x_{n-k-2}} u_{x_{n-k-1}}
 \ \! \lambda ( dx_1 ) \cdots \lambda ( dx_{n-k-1} )
\\
 & =
 \sum_{i=1}^{k+1}
 \frac{m!}{(m+i)!}
 ( \widetilde{\nabla}^* u )^{n-i}
 D_{x_0}
 \int_{B(R)}
 u_s^{m+i}
 \ \! \lambda ( ds )
 +
 \frac{m!}{(m+k+1)!}
 ( \widetilde{\nabla}^* u )^{n-k-1}
 u_{x_0}^{m+k+1}
,
\end{align*}
which shows by induction that \eqref{id} holds
at the rank $k=n$, in particular we have
\begin{equation}
\nonumber
 ( \widetilde{\nabla}^* u )^n u^m_x
 =
 \frac{m!}{(m+k)!}
 u^{m+n}_x
 +
 \sum_{i=2}^{n+1}
 \frac{m!}{(m+i-1)!}
 ( \widetilde{\nabla}^* u )^{n+1-i}
 D_x
 \int_{B(R)}
 u_y^{m+i-1}
 \ \! \lambda ( dy )
,
\end{equation}
$x \in B(R)$, which yields \eqref{djklll}
by integration with respect to $x \in B(R)$ and duality.
\end{Proof}
 As a consequence of Lemma~\ref{itlw2.1} we have
\begin{equation}
\nonumber
 \Gamma^u_k {\bf 1}
 =
 \int_{B(R)}
 \frac{u_x^k }{(k-1)!} \ \! \lambda ( dx )
 +
 \sum_{i=2}^{k-1}
 \frac{1}{i!}
 \left<
 ( \widetilde{\nabla} u )^{k-1-i}
 u ,
 D
 \int_{B(R)}
 u^i_x \ \! \lambda ( dx )
 \right>_{L^2(B(R))},
\end{equation}
$k\geq 2$.
Hence when $h\in W^{1,p}( B(R) )$, $p>1$,
is a deterministic
function such that $\Vert \nabla^{\real^d} h\Vert_\infty < \infty$,
 we find the relation
\begin{equation}
\label{ifinad2}
 \Gamma^h_k {\bf 1}
 =
 \frac{1}{(k-1)!}
 \int_{B(R)}
 h^k(x) \ \! \lambda ( dx )
 =
 \frac{1}{(k-1)!}
 \kappa_k^h
 , \qquad
 k \geq 2,
\end{equation}
 which shows that $\Gamma^h_k {\bf 1}$ coincides with
 the cumulant $\kappa_k^h = \int_{B(R)} h^k(x) \ \! \lambda ( dx )$
 of order $k\geq 2$ of the Poisson stochastic
 integral $\int_{B(R)} h(x) ( \gamma ( dx ) - \lambda (dx) )$.
\section{Edgeworth type expansions}
\label{s3}
 Classical Edgeworth series provide expansion
 of the cumulative distribution function $P(F\leq x)$
 of a centered random variable $F$ with $E[F^2]=1$
 around the Gaussian cumulative distribution function
 $\Phi (x)$, using the cumulants $(\kappa_n)_{n\geq 1}$ of a random variable $F$
 and Hermite polynomials.
 Edgeworth type expansions of the form
\begin{equation}
\nonumber
 E[Fg(F) ] = \sum_{l=1}^n \frac{\kappa_{l+1}}{l!} E [ g^{(l)} (F) ]
 + E[g^{(n+1)} (F) \Gamma_{n+1} F ], \quad n \geq 1,
\end{equation}
 for $F$ a centered random variable,
 have been obtained by the Malliavin calculus in
 \cite{nourdinpeccati2009},
 where $\Gamma_{n+1}$ is a cumulant type
 operator on the Wiener space such that
 $n! E[\Gamma_n F ]$ coincides with the cumulant
 $\kappa_{n+1}$ of order $n+1$ of $F$, $n\in \inte$, cf.
 \cite{nourdin10}, extending the results of \cite{barbour1986}
 to the Wiener space.
\\

In this section we establish an Edgeworth type expansion
of any finite order with an explicit remainder term for the
compensated Poisson stochastic integral $\delta (u)$ of
a
predictable random field $(u_x)_{x\in B(R)}$.
In the sequel we let $\langle \cdot , \cdot \rangle$
denote $\langle \cdot , \cdot \rangle_{L^2(B(R))}$, except
if stated otherwise.
\\

 Before proceeding to the statement of
 general expansions in Proposition~\ref{djklsad1},
 we illustrate the method with the derivation of an expansion of order one for
 a deterministic integrand $f$.
 By the duality relation \eqref{jldf} between $D$ and $\delta$,
 the chain rule of derivation for $D$ and
 the commutation relation \eqref{com} we get,
 for ${{g}} \in {\cal C}^2_b(\real )$ and
 $f\in W^{1,1}_0(B(R))$ such that
 $\Vert \nabla^{\real^d} f\Vert_\infty < \infty$,
\begin{eqnarray*}
  \lefteqn{
    E [
 \delta ( f )
 {{g}} ( \delta ( f ) )
 ]
 =
 E [
 \langle f , D \delta ( f ) \rangle
         {{g}}' ( \delta ( f ) )
   ]
  }
  \\
 & = &
 E[
 \langle
 f ,
 f
 \rangle
 g' ( \delta ( f ) )
 ]
 +
 E \big[
   \langle f , \delta ( \widetilde{\nabla}^* f) \rangle
   g' ( \delta (f) )
 \big]
 \\
 & = &
 E[
 \langle
 f ,
 f
 \rangle
 g' ( \delta ( f ) )
 ]
+
 E \big[
 \langle \widetilde{\nabla}^* f ,
 D (
 g' ( \delta (f) )
 f
 )
 \rangle
 \big]
 \\
 & = &
 E[
 \langle
 f ,
 f
 \rangle
 g' ( \delta ( f ) )
 ]
 +
  E\big[
 \langle
 ( \widetilde{\nabla} f )
 f
 ,
 D
 \delta ( f )
 \rangle
 g'' ( \delta ( f ) )
 \big]
\\
 & = &
 E[
 \langle
 f ,
 f
 \rangle
 g' ( \delta ( f ) )
 ]
 +
\frac{1}{2}
\int_{B(R)} f^3(x) \ \! \lambda ( dx )
 E[
 g'' ( \delta ( f ) )
 ]
 +
 E\big[
 \langle
 ( \widetilde{\nabla} f )
 f
 ,
 \delta ( \widetilde{\nabla}^* f )
 \rangle
 g'' ( \delta ( f ) )
 \big]
\\
 & = &
\kappa_2^f
 E[
 g' ( \delta ( f ) )
 ]
 +
\frac{1}{2}
\kappa_3^f
 E[
 g'' ( \delta ( f ) )
 ]
 +
 E\big[
 g'' ( \delta ( f ) )
 \delta (
 ( \widetilde{\nabla} f )^2 f
 )
 \big]
,
\end{eqnarray*}
 since by Lemma~\ref{itlw2.1} we have
$$
\langle (\widetilde{\nabla} f)f , f\rangle
 =
\frac{1}{2}
\int_{B(R)} f^3(x) \ \! \lambda ( dx )
=
\frac{1}{2}
\kappa_3^f.
$$
In the next proposition we derive general Edgeworth type
expansions for predictable integrand processes $(u_x)_{x \in \real^d}$.
\begin{prop}
\label{djklsad1}
Let $u \in \widetilde{\dee}^{1,\infty}_0$ and $n \geq 0$.
For all ${{g}} \in {\cal C}^{n+1}_b(\real )$
 and bounded $G\in \dee_{1,1}$ we have
\begin{align}
\nonumber
 &   E\left[
 G
 \delta ( u )
 {{g}} ( \delta ( u ) )
 \right]
  =
 E\left[
 \langle u , DG\rangle
 g ( \delta ( u ) )
 \right]
 +
 \sum_{k=1}^n
 E\left[
 {{g}}^{(k)} ( \delta ( u ) )
 \Gamma_{k+1}^u G
 \right]
\\
\nonumber
 &
+
 E\left[
 G
       {{g}}^{(n+1)} ( \delta ( u ) )
 \left(
 \int_{B(R)}
 \frac{ u_x^{n+2} }{(n+1)!}
 \ \! \lambda ( dx )
 + \sum_{k=2}^{n+1}
 \left<
 ( \widetilde{\nabla} u )^{n+1-k}
 u ,
 D \int_{B(R)} \frac{u^k_x}{k!} \ \! \lambda ( dx )
 \right>
 \right)
       \right]
\\
\nonumber
 &
+
 E\left[
 G
       {{g}}^{(n+1)} ( \delta ( u ) )
 \langle
 ( \widetilde{\nabla} u )^n
 u
 ,
 \delta ( \widetilde{\nabla}^* u )
 \rangle
       \right]
.
\end{align}
\end{prop}
\begin{Proof}
 By the duality relation \eqref{jldf} between $D$ and $\delta$,
 the chain rule of derivation for $D$ and
 the commutation relation \eqref{com}, we get
\begin{align*}
\nonumber
 & E\big[
 G
 \langle
 ( \widetilde{\nabla} u )^k
 u ,
 D
 \delta ( u )
 \rangle
 {{g}} ( \delta ( u ) )
 \big]
 -
 E\big[
 G \langle
 ( \widetilde{\nabla} u )^{k+1}
 u
 ,
 D
 \delta ( u )
 \rangle
 {{g}}' ( \delta ( u ) )
 \big]
\\
 & =
 E\big[
 G \langle
 ( \widetilde{\nabla} u )^k
 u ,
 u
 \rangle
 {{g}} ( \delta ( u ) )
 \big]
 +
 E \big[ G
   \langle ( \widetilde{\nabla} u)^k u , \delta ( \widetilde{\nabla}^* u) \rangle
           {{g}} ( \delta (u) )
 \big]
  -
 E\big[
 G \langle
 ( \widetilde{\nabla} u )^{k+1}
 u
 ,
 D
 \delta ( u )
 \rangle
 {{g}}' ( \delta ( u ) )
 \big]
 \\
 & =
 E\big[
 G \langle
 ( \widetilde{\nabla} u )^k
 u ,
 u
 \rangle
 {{g}} ( \delta ( u ) )
 \big]
+
 E \big[
 \langle \widetilde{\nabla}^* u ,
 D (
 G
 {{g}} ( \delta (u) )
 ( \widetilde{\nabla} u)^k u
 )
 \big]
 \rangle
 - E\big[
 G \langle
 ( \widetilde{\nabla} u )^{k+1}
 u
 ,
 D
 \delta ( u )
 \rangle
 {{g}}' ( \delta ( u ) )
 \big]
\\
 & =
 E\big[
 G \langle
 ( \widetilde{\nabla} u )^k
 u ,
 u
 \rangle
 {{g}} ( \delta ( u ) )
 \big]
 +
 E \big[
 \langle (\widetilde{\nabla} u)^{k+1} u ,
 D G \rangle
 {{g}} ( \delta (u) )
 \big]
 +
 E \big[
   G
 \langle \widetilde{\nabla}^* u ,
 D ( ( \widetilde{\nabla} u)^k u )
 \rangle
   {{g}} ( \delta (u) )
 \big]
\\
 & =
 E\big[
 {{g}} ( \delta ( u ) )
 \Gamma_{k+2}^u G
 \big],
\end{align*}
where we used \eqref{comp2} and \eqref{comp3}.
 Therefore, we have
\begin{align*}
 &
 E[
 G
 \delta ( u )
 {{g}} ( \delta ( u ) )
 ]
 =
 E[
   \langle u , D ( G {{g}} ( \delta ( u ) ) )
   \rangle
 ]
\\
 & =
 E[
 G
 \langle u , D \delta ( u ) \rangle
 {{g}}' ( \delta ( u ) )
 ]
 +
 E[
 \langle u , D G \rangle
 {{g}} ( \delta ( u ) )
 ]
\\
 & =
 E[
 \langle u , D G \rangle
 {{g}} ( \delta ( u ) )
 ]
 +
 E\big[
 G  {{g}}^{(n+1)} ( \delta ( u ) )
\langle
 ( \widetilde{\nabla} u )^n
 u
 ,
 D
 \delta ( u )
 \rangle
 \big]
\\
 &
 \quad +
 \sum_{k=0}^{n-1}
 \big(
 E\big[
 G
 {{g}}^{(k+1)} ( \delta ( u ) )
 \langle
 ( \widetilde{\nabla} u )^k
 u ,
 D
 \delta ( u )
 \rangle
 \big]
 -
 E\big[
 G  {{g}}^{(k+2)} ( \delta ( u ) )
\langle
 ( \widetilde{\nabla} u )^{k+1}
 u
 ,
 D
 \delta ( u )
 \rangle
 \big]
 \big)
\\
 & =
 E[
 \langle u , D G \rangle
 {{g}} ( \delta ( u ) )
 ]
 +
 \sum_{k=1}^n
 E\big[
 {{g}}^{(k)} ( \delta ( u ) )
 \Gamma_{k+1}^u G
 \big]
 +
 E\big[
 G  {{g}}^{(n+1)} ( \delta ( u ) )
\langle
 ( \widetilde{\nabla} u )^n
 u
 ,
 D
 \delta ( u )
 \rangle
 \big]
\\
 & =
 E[
 \langle u , D G \rangle
 {{g}} ( \delta ( u ) )
 ]
 +
 \sum_{k=1}^n
 E\big[
 {{g}}^{(k)} ( \delta ( u ) )
 \Gamma_{k+1}^u G
 \big]
 \\
 & \quad
 +
 E\big[
 G  {{g}}^{(n+1)} ( \delta ( u ) )
\langle ( \widetilde{\nabla} u )^n u , u \rangle
 \big]
 +
 E\big[
 G  {{g}}^{(n+1)} ( \delta ( u ) )
\langle
 ( \widetilde{\nabla} u )^n
 u
 ,
 \delta ( \widetilde{\nabla}^* u )
 \rangle
 \big],
\end{align*}
 and we conclude by Lemma~\ref{itlw2.1}.
\end{Proof}
When $f\in W^{1,1}_0(B(R))$
is a deterministic function such that
$\Vert \nabla^{\real^d} f\Vert_\infty < \infty$, and $g\in {\cal C}^\infty_b ( \real )$,
Proposition~\ref{djklsad1} shows that
\begin{eqnarray*}
  \lefteqn{
    E\left[
 \delta ( f )
 {{g}} ( \delta ( f ) )
 \right]
  }
  \\
  & = &
 \sum_{k=1}^{n+1}
 \frac{1}{k!}
 \int_{B(R)} f^{k+1}(x) \ \! \lambda ( dx )
 E[
 {{g}}^{(k)} ( \delta ( f ) )
 ]
 + E\big[
 {{g}}^{(n+1)} ( \delta ( f ) )
 \langle
 ( \widetilde{\nabla} f )^n
 f
 ,
 \delta ( \widetilde{\nabla}^* f )
 \rangle
 \big]
 \\
  & = &
 \sum_{k=1}^{n+1}
 \frac{1}{k!}
 \kappa_{k+1}^f
 E[
 {{g}}^{(k)} ( \delta ( f ) )
 ]
 + E\big[
 {{g}}^{(n+1)} ( \delta ( f ) )
 \delta ( ( \widetilde{\nabla} f )^{n+1} f )
 \big], \qquad n \geq 0,
\end{eqnarray*}
 with, by Proposition~\ref{p1} applied with $p=2$ and $r=0$,
\begin{eqnarray*}
  E\big[
 |
 \delta (
  ( \widetilde{\nabla} f )^{n+1}
 f
 )
 |
 \big]
 & \leq &
 \sqrt{E\big[
 |
 \delta ( ( \widetilde{\nabla} f )^{n+1}
 f
 )
 |^2
 \big]}
\\
 & = &
 \Vert
 ( \widetilde{\nabla} f )^{n+1} f
 \Vert_{L^2(B(R))}
\\
 & \leq &
  ( K_d v_d R' )^{n+1}
  \Vert f \Vert_{L^2 (B(R))}
  \Vert \widetilde{\nabla} f \Vert_{L^\infty (B(R);\real^d)}^{n+1}.
\end{eqnarray*}
 In addition, as $n$ tends to $+\infty$ we have
\begin{eqnarray*}
 E\left[
 \delta ( f )
 {{g}} ( \delta ( f ) )
 \right]
 & = &
 \sum_{k=1}^\infty
  \frac{1}{k!}
 \int_{B(R)}
f^{k+1}(x)
 \ \! \lambda ( dx )
 E\left[
 {{g}}^{(k)} ( \delta ( f ) )
 \right]
\\
\nonumber
& = &
 \sum_{k=1}^\infty
 \frac{1}{k!}
 \int_{B(R)}
 f^{k+1}(x) \ \! \lambda ( dx )
 E\left[
 {{g}}^{(k)} ( \delta ( f ) )
 \right]
 \\
  & = & E\left[
 \int_{B(R)} f(x)
 \big( {{g}} ( \delta ( f ) + f(x) ) - {{g}} ( \delta ( f ) ) \big)
 \lambda ( dx )
 \right]
\end{eqnarray*}
 provided that the derivatives of $g$ decay fast enough,
 which is a particular instance of
the standard integration by parts identity
for finite difference operators on the Poisson space,
see e.g. Lemma~2.9 in \cite{utzet2} or Lemma~5 in \cite{bourguin}.
\section{Stein approximation}
\label{s4}
Applying Proposition~\ref{djklsad1}
with $n=0$ and $G=1$ to the solution $g_x$ of the Stein equation
\begin{equation}
\nonumber
 {\bf 1}_{(-\infty , x ]}(z) - \Phi ( z ) = g'_x ( z ) - z g_x (z), \qquad z\in\real,
\end{equation}
 and $u \in \widetilde{\dee}^{1,1}_0$ a predictable random field
 this gives the expansion
\begin{eqnarray}
\nonumber
 P ( \delta ( u ) \leq x )
 -
 \Phi ( x )
 & = &
 E\big[ {{g}}'_x ( \delta ( u ) )
 \langle
 u
 ,
 u \rangle
 -
 \delta ( u )
 {{g}}_x ( \delta ( u ) )
 \big]
 \\
 \nonumber
 & = &
 E\left[
 ( 1 - \langle
 u
 ,
 u \rangle
 ) {{g}}'_x ( \delta ( u ) )
\right]
 +
 E\big[
 \langle u , \delta ( \widetilde{\nabla} u ) \rangle
 {{g}}'_x ( \delta ( u ) )
 \big],
\end{eqnarray}
 around the Gaussian cumulative distribution function $\Phi (x)$,
 with $\Vert g_x \Vert_\infty \leq \sqrt{2\pi}/4$
 and $\Vert g'_x \Vert_\infty \leq 1$,
 $x\in \real$, by Lemma~2.2-$(v)$ of \cite{chenbk}.
 The next result applies Proposition~\ref{djklsad1}
 with $n=1$ and $G=1$.
\begin{prop}
\label{djhlkdsa}
For any random field $u \in \widetilde{\dee}^{1,\infty}_0$
 we have
\begin{eqnarray}
\nonumber
\lefteqn{
  d ( \delta ( u ) , {\cal N} )
}
\\
\nonumber
& \leq &
 E\big[
 | 1 -  \langle
 u
 ,
 u
 \rangle
 -  \langle \widetilde{\nabla}^* u , D u \rangle
 |
 \big]
+
   E\left[
   \left|
   \int_{B(R)} u_x^3 \ \! \lambda ( dx )
 +
\left<
 u ,
 D
 \int_{B(R)} u^2_x \ \! \lambda ( dx )
 \right>
 \right|
 \right]
\\
\label{djklsadsa11}
  & &
 +
 2 E\big[
 |
 \langle
 ( \widetilde{\nabla} u )
 u
 ,
 \delta ( \widetilde{\nabla}^* u )
 \rangle
 |
 \big]
.
\end{eqnarray}
\end{prop}
\begin{Proof}
 For $n=1$ and $G=1$, Proposition~\ref{djklsad1} shows that
\begin{eqnarray*}
\nonumber
 E [ \delta ( u )
 {{g}} ( \delta ( u ) )
 ]
 & = &
 E [
 {{g}}' ( \delta ( u ) )
 ( \langle u , u \rangle
 + \langle \widetilde{\nabla}^* u , D u \rangle
 )
 ]
 \\
 & &
 +
 \frac{1}{2}
 E\left[
 g'' ( \delta ( u ) )
 \left(
 \int_{B(R)} u_x^3 \ \! \lambda ( dx )
 +
 \left<
 u ,
 D
 \int_{B(R)} u^2_x \ \! \lambda ( dx )
 \right>
 \right)
 \right]
\\
 & &
 +
 E [
 {{g}}'' ( \delta ( u ) )
 \langle
 ( \widetilde{\nabla} u )
 u
 ,
 \delta ( \widetilde{\nabla} u )
 \rangle
 ]
.
\end{eqnarray*}
Let $h:\real\to [0,1]$
be a continuous function
with bounded derivative.
Using the solution $g_h\in\mathcal{C}_b^1(\real)$
 of the Stein equation
\begin{equation}
\nonumber
 h( z )-\mathrm{E}[h( {\cal N}  )] = g' ( z )- z g( z ), \qquad z\in\real,
\end{equation}
 with the bounds
 $\Vert g'_h \Vert_\infty \leq \Vert h' \Vert_\infty$
 and
 $\Vert g''_h \Vert_\infty
 \leq 2 \Vert h' \Vert_\infty$,
 $x \in \real$,
 cf. Lemma~1.2-$(v)$ of \cite{nourdinpeccati} and references therein,
 we have
\begin{eqnarray*}
\nonumber
 E [ h ( \delta ( u ) ) ]
 -
 E [ h ( {\cal N} ) ]
 & = &
 E [
 \delta ( u )
 g_h ( \delta ( u ) )
 -
 g_h' ( \delta ( u ) )
 ]
\\
 & = &
 E [
 g_h' ( \delta ( u ) )
 ( \langle u , u \rangle
 +  \langle \widetilde{\nabla}^* u , D u \rangle
 - 1 )
 ]
 \\
  & & +
 \frac{1}{2}
 E\left[
 g'' ( \delta ( u ) )
 \left(
 \int_{B(R)} u_x^3 \ \! \lambda ( dx )
 + \left<
 u ,
 D
 \int_{B(R)} u^2_x \ \! \lambda ( dx )
 \right>
 \right) \right]
 \\
 & &
 +
 2 E[
 g_h'' ( \delta ( u ) )
 \langle
 ( \widetilde{\nabla} u )
 u
 ,
 \delta ( \widetilde{\nabla}^* u )
 \rangle
 ]
,
\end{eqnarray*}
 hence
\begin{eqnarray*}
\nonumber
  | E [ \delta ( u )
 h ( \delta ( u ) )
 ]
 -
 E [
 h ( {\cal N} )
 ] |
& \leq &
 \Vert h' \Vert_\infty
 E\big[
   | 1 - \langle u , u \rangle
   -  \langle \widetilde{\nabla}^* u , D u \rangle
|
 \big]
 \\
   &  &
 +
 \Vert h' \Vert_\infty
 E\left[
   \left|
   \int_{B(R)} u_x^3 \ \! \lambda ( dx )
 +
\left<
 u ,
 D
 \int_{B(R)} u^2_x \ \! \lambda ( dx )
 \right>
 \right|
 \right]
 \\
   &  &
   +
 2 \Vert h' \Vert_\infty
 E\big[
 |
 \langle
 ( \widetilde{\nabla} u )
 u
 ,
 \delta ( \widetilde{\nabla}^* u )
 \rangle
 |
 \big]
,
\end{eqnarray*}
which yields \eqref{djklsadsa11}.
\end{Proof}
 As a consequence of Proposition~\ref{djhlkdsa}
 and the It\^o isometry \eqref{isomrel} we have the following corollary.
\begin{corollary}
\label{djkd}
For $u \in \widetilde{\dee}^{1,\infty}_0$
 we have
\begin{eqnarray}
 \nonumber
 d ( \delta ( u ) , {\cal N} )
 & \leq &
 | 1 - \Var [ \delta (u) ] |
 +
 \sqrt{
 \Var \big[
 \Vert u \Vert_{L^2(B(R))}^2
 \big]
 }
 \\
 \nonumber
 & &
 +
   E\left[
   \left|
   \int_{B(R)} u_x^3 \ \! \lambda ( dx )
 +
\left<
 u ,
 D
 \int_{B(R)} u^2_x \ \! \lambda ( dx )
 \right>
 \right|
 \right]
\\
\nonumber
 & &
+ E[| \langle \widetilde{\nabla}^* u , D u \rangle
  | ]
+
 2 E\big[
 |
 \langle
 ( \widetilde{\nabla} u )
 u
 ,
 \delta ( \widetilde{\nabla}^* u )
 \rangle
 |
 \big]
.
\end{eqnarray}
\end{corollary}
\begin{Proof}
  By the It\^o isometry \eqref{isomrel} we have
$$
  \Var [ \delta (u) ] = E \left[ \left(
    \int_{B(R)} u_x ( \gamma ( dx ) - \lambda (dx) ) \right)^2 \right]
 =
 E [
 \langle
 u
 ,
 u
 \rangle
 ],
 $$
  hence
\begin{eqnarray*}
\nonumber
\lefteqn{
  E\big[
 | 1 -  \langle
 u
 ,
 u
 \rangle
-  \langle \widetilde{\nabla}^* u , D u \rangle
 |
 \big]
}
\\
& \leq &
 E\left[
 | 1  -
 E [ \langle
 u
 ,
 u
 \rangle]
 |
 \right]
 +
   E\left[
 | \langle
 u
 ,
 u
 \rangle
 -
 E [ \langle
 u
 ,
 u
 \rangle]
 |
 \right]
   + E[| \langle \widetilde{\nabla}^* u , D u \rangle
     | ]
\\
 & = &
 | 1 - \Var [ \delta ( u ) ] |
 +
 \sqrt{E [ ( \langle u , u \rangle
     - E[\langle u , u \rangle
     ] )^2 ]}
 + E[| \langle \widetilde{\nabla}^* u , D u \rangle
   | ]
 \\
  & = &
 | 1 - \Var [ \delta ( u ) ] |
 +
  \sqrt{\Var \big[ \Vert u \Vert_{L^2(B(R))}^2 \big] }
  + E[| \langle \widetilde{\nabla}^* u , D u \rangle
    | ]
.
\end{eqnarray*}
\end{Proof}
In particular, when $\Var [ \delta (u) ] = 1$,
Corollary~\ref{djkd}
shows that
\begin{eqnarray*}
 d ( \delta ( u ) , {\cal N} )
 & \leq &
 \sqrt{
 \Var \big[
 \Vert u \Vert_{L^2(B(R))}^2
 \big] }
 +
   E\left[
   \left|
   \int_{B(R)} u_x^3 \ \! \lambda ( dx )
 +
\left<
 u ,
 D \int_{B(R)} u^2_x \ \! \lambda ( dx )
 \right>
 \right|
 \right]
   \\
   & &
   + E[| \langle \widetilde{\nabla}^* u , D u \rangle
     | ]
 +
 2 E\big[
 |
 \langle
 ( \widetilde{\nabla} u )
 u
 ,
 \delta ( \widetilde{\nabla}^* u )
 \rangle
 |
 \big]
.
\end{eqnarray*}
 When $f\in W^{1,\infty}_0(B(R))$
is a deterministic function we have
$$
 \Var [ \delta (f) ]
=
\E \left[
  \left(
  \int_{B(R)} f(x) ( \gamma ( dx ) - \lambda (dx) )
  \right)^2
  \right]
=
\int_{B(R)} f^2(x) \ \! \lambda ( dx ),
$$
 and Corollary~\ref{djhlkdsa} shows that
$$
 d ( \delta ( f ) , {\cal N} )
 \leq
   \left| 1 - \int_{B(R)} f^2(x) \ \! \lambda (dx)
 \right|
  +
  \left|
  \int_{B(R)} f^3(x) \ \! \lambda ( dx )
 \right|
 +
2  E\big[
 |
 \delta ( ( \widetilde{\nabla} f )^2 f )
 |
 \big]
.
$$
 Given the bound
\begin{eqnarray*}
  E\big[
 |
 \delta (
  ( \widetilde{\nabla} f )^2
 f
 )
 |
 \big]
 & \leq &
 \sqrt{E\big[
 |
 \delta (
  ( \widetilde{\nabla} f )^2
 f
 )
 |^2
 \big]}
\\
 & = &
 \Vert
   ( \widetilde{\nabla} f )^2
 f
 \Vert_{L^2(B(R))}
\\
 & \leq &
  ( K_d v_d R' )^2
  \Vert f \Vert_{L^2 (B(R))}
  \Vert \nabla^{\real^d} f \Vert_{L^\infty (B(R);\real^d)}^2
\end{eqnarray*}
obtained from Proposition~\ref{p1} with $p=2$ and $r=0$,
$f\in W^{1,\infty}_0(B(R))$, we have the following corollary.
\begin{corollary}
 For $f\in W^{1,\infty}_0(B(R))$ we have
\begin{eqnarray}
\nonumber
 d \left( \int_{B(R)} f(x) ( \gamma ( dx ) - \lambda (dx) ), {\cal N} \right)
 & \leq &
   \big| 1 - \Vert f\Vert_{L^2(B(R))}^2
 \big|
+ \left| \int_{B(R)} f^3(x) \ \! \lambda ( dx ) \right|
\\
\nonumber
& &
+ 2 ( K_d v_d R' )^2
  \Vert f \Vert_{L^2 (B(R))}
  \Vert \nabla^{\real^d} f \Vert_{L^\infty (B(R);\real^d)}^2.
\end{eqnarray}
\end{corollary}
In particular, if $\Vert f\Vert_{L^2(B(R))}=1$ we find
$$
 d \left( \int_{B(R)} f(x) ( \gamma ( dx ) - \lambda (dx) ), {\cal N} \right)
 \leq
 \left| \int_{B(R)} f^3(x) \ \! \lambda ( dx ) \right|
+ 2 ( K_d v_d R' )^2
\Vert \nabla^{\real^d} f \Vert_{L^\infty (B(R);\real^d)}^2.
$$
As an example, consider $f_k$ given on
$B(Rk^{1/d})$ by
$$
f_k (x) := \frac{1}{C \sqrt{k} } g\left(\frac{|x|_{\real^d}}{k^{1/d}}\right)
,
$$
where $g\in {\cal C}^1_0 ([0,R])$
and
$$
C^2 := v_d \int_0^R g^2(r) r^{d-1} dr,
$$
so that
$f_k \in L^2(B(R k^{1/d}))$ with
$$
 \Vert f \Vert_{L^2(B(R k^{1/d}))}^2
=
\frac{v_d}{C^2 k}
\int_0^{ R k^{1/d}} g^2 \left( \frac{r}{k^{1/d}} \right) r^{d-1} dr
=
\frac{v_d}{C^2} \int_0^R g^2 (r) r^{d-1} dr
=
1,
$$
 and
$$
\int_{B(Rk^{1/d})} f_k^3 (x) dx
=
\frac{1}{C^3 k^{3/2}}
\int_0^{Rk^{1/d}} g^3 (r k^{-1/d}) r^{d-1} dr
=
\frac{1}{C^3\sqrt{k}}
\int_0^R g^3 (r) r^{d-1} dr,
$$
 $k \geq 1$.
We have
$$
\Vert \nabla^{\real^d} f_k \Vert_{L^\infty (B(R);\real^d)}^2 \leq
\frac{\Vert g' \Vert_\infty^2 d}{C^2 k^{1+2/d}},
$$
 hence
\begin{eqnarray*}
 d \left( \int_{B(R)} f_k(x) ( \gamma ( dx ) - \lambda (dx) ), {\cal N} \right)
 & \leq &
 \left| \int_{B(R)} f^3_k(x) \ \! \lambda ( dx ) \right|
 + \frac{ 2 ( K_d v_d k^{1/d} R' )^2 d }{ k^{1+2/d} C^2}\Vert g' \Vert_\infty^2
\\
 & \leq &
\frac{v_d}{C^3\sqrt{k}}
\left| \int_0^R g^3 (r) r^{d-1} dr \right|
+ \frac{ 2 ( K_d v_d R' )^2 d }{ k C^2}\Vert g' \Vert_\infty^2.
\end{eqnarray*}
In particular, if $g$ satisfies the condition
$$
\int_0^R g^3 (r) r^{d-1} dr =0,
$$
 then we find the $O(1/k)$ convergence rate
$$
 d \left( \int_{B(R)} f_k(x) ( \gamma ( dx ) - \lambda (dx) ), {\cal N} \right)
 \leq \frac{ 2 ( K_d v_d R' )^2 d}{ k C^2},
 \qquad k \geq 1.
$$
 \section{Appendix}
 \begin{Proofy} \hskip-0.32cm {\em of Proposition~\ref{prop1}.}

 As a consequence of \eqref{gxy4} and \eqref{cont} we have
\begin{eqnarray}
\nonumber
\lefteqn{
 f_n \left( x_1,\ldots , x_{i-1}, y , x_{i+1}, \ldots x_n \right)
 -
 f_{n-1} \left( x_1,\ldots , x_{i-1},x_{i+1}, \ldots, x_n \right)
}
\\
\nonumber
& = &
 f_n \left( x_1,\ldots , x_{i-1}, y , x_{i+1}, \ldots x_n \right)
 -
 f_{n-1} \left( x_1,\ldots , x_{i-1},x_{i+1}, \ldots, x_n \right)
 \int_{B(R')\setminus B(R)}
 \eta(x)
 \ \! \lambda (dx)
 \\
\nonumber
  & = &
 f_n \left( x_1,\ldots , x_{i-1}, y , x_{i+1}, \ldots x_n \right)
 -
 \int_{B(R')\setminus B(R)}
 \eta(x)
 f_n \left( x_1,\ldots , x_{i-1},x,x_{i+1}, \ldots, x_n \right)
 \ \! \lambda (dx)
 \\
\nonumber
  & = &
\int_{B(R')} \langle G(x_i,y)
,
\nabla^{\real^d}_{x_i} f_n \left( x_1,\ldots , x_n \right)
\rangle_{\real^d}
\lambda ( dx_i )
 \\
\label{cont2}
  & = &
\int_{B(R)} \langle G(x_i,y)
,
\nabla^{\real^d}_{x_i} f_n \left( x_1,\ldots , x_n \right)
\rangle_{\real^d}
\lambda ( dx_i )
,
\end{eqnarray}
$x_1,\ldots , x_{i-1},y,x_{i+1}, \ldots , x_n\in B(R')$.
 Recall that for all $F\in {\mathcal S}$ of the form \eqref{f} we have
$$
 E [F] = e^{-B(R)}
 \sum_{n=0}^\infty \frac{1}{n!}
 \int_{B(R)} \cdots \int_{B(R)} f_n(x_1,\ldots ,x_n) \ \! \lambda ( dx_1 ) \cdots \lambda (dx_n).
$$
 Hence, using \eqref{cont2},
 for $g\in {\cal C}^1_0 (B(R))$ and $F$ of the form \eqref{f}
 we have
  \begin{align}
  \nonumber
  &
  E \left[ \int_{B(R)} g(y) D_y F \ \! \lambda ( dy ) \right]
 \\
  & =
 E\left[
   \sum_{n=1}^\infty
   {\bf 1}_{\{ \gamma ( B(R))=n\} }
 \sum_{i=1}^n
\int_{B(R)}
 g(y) \langle \G_\eta (X_i,y) ,
 \nabla^{\real^d}_{X_i} f\left( X_1,\ldots , X_n \right)
 \rangle \lambda ( dy)
 \right]
    \\
  \nonumber
  & =
  e^{-B(R)}
 \sum_{n=1}^\infty \frac{1}{n!}
 \int_{B(R)} \cdots \int_{B(R)}
 \sum_{i=1}^n
\int_{B(R)}
 g(y) \langle \G_\eta (x_i,y) ,
 \nabla^{\real^d}_{x_i} f_n(x_1,\ldots ,x_n)
 \rangle \lambda ( dy)
   \lambda ( dx_1 ) \cdots \lambda (dx_n)
    \\
  \nonumber
  & =
  e^{-B(R)}
  \sum_{n=1}^\infty
  \frac{1}{n!}
  \int_{B(R)} \cdots
  \\
  \nonumber
  & \quad
\cdots \int_{B(R)}
 \sum_{i=1}^n
\int_{B(R)}
g(y)
 f_n(x_1,\ldots ,x_{i-1},y,x_{i+1},\ldots , x_n)
   \ \! \lambda ( dx_1 ) \cdots  \lambda ( dy) \cdots \lambda (dx_n)
   \\
   \nonumber
        &
 \quad -
  e^{-B(R)}
 \sum_{n=1}^\infty \frac{1}{n!}
 \int_{B(R)} \cdots \int_{B(R)}
 \sum_{i=1}^n
\int_{B(R)}
g(y)
 \ \! \lambda ( dy)
 f_{n-1}(x_1,\ldots ,x_{n-1})
   \ \! \lambda ( dx_1 ) \cdots \lambda (dx_{n-1})
    \\
  \nonumber
  & =
  e^{-B(R)}
  \sum_{n=1}^\infty
  \frac{1}{n!}
  \int_{B(R)} \cdots \int_{B(R)}
  \left(
  \sum_{i=1}^n
g(x_i)
- \int_{B(R)}
g(y)
 \ \! \lambda ( dy)
 \right)
 f_n (x_1,\ldots ,x_n )
 \ \! \lambda ( dx_1 ) \cdots \lambda (dx_n)
    \\
\nonumber
  & =
    E\left[
      F
      \left(
      \int_{B(R)} g(x) ( \gamma ( dx ) - \lambda (dx) )
      \right)
      \right].
\end{align}
Next, for $u$ of the form \eqref{u}, we check by a standard argument that
\begin{eqnarray*}
  \nonumber
E[    \langle u , DF\rangle_{L^2(B(R))} ]
  & = &
\sum_{i=1}^n
E[ G_i \langle g_i , DF\rangle_{L^2(B(R))} ]
\\
& = &
\sum_{i=1}^n
\left(
E[ \langle g_i , D( F G_i ) \rangle_{L^2(B(R))} - F \langle g_i , DG_i \rangle_{L^2(B(R))} ]
\right)
\\
& = &
    E\left[
      F
\sum_{i=1}^n
\Big(
G_i
     \int_{B(R)} g_i(x) ( \gamma ( dx ) - \lambda (dx) )
-
\langle g_i , DG_i \rangle_{L^2(B(R))}
\Big)
    \right]
\\
& = &
    E [
      F \delta ( u )].
    \end{eqnarray*}
\end{Proofy}
\begin{Proofy} \hskip-0.32cm {\em of Proposition~\ref{prop2}.}
 Taking $u \in {\cal P}_0$ a predictable random field
of the form \eqref{uf} we note that by \eqref{ndfgd} and the compatibility condition
\eqref{compat} we have
$$
g_i(y)D_yF_i=0, \qquad y\in B(R), \quad i=1,\ldots , n,
$$
hence by
\eqref{dltu} we have
\begin{eqnarray}
  \label{dltu2}
\delta ( u )
& = & \delta \left( \sum_{i=1}^n F_i g_i \right)
 = \sum_{i=1}^n F_i \delta ( g_i )
\\
\nonumber
& = & \sum_{i=1}^n F_i \int_{B(R)} g_i(x) ( \gamma ( dx ) - \lambda ( dx ))
\\
\nonumber
& = & \int_{B(R)} u_x ( \gamma ( dx ) - \lambda ( dx )),
\end{eqnarray}
 showing that $\delta (u)$ coincides with the
 Poisson stochastic integral of $(u_x)_{x\in B(R)}$.
 Regarding the isometry relation \eqref{isomrel}, we have
\begin{eqnarray*}
      E [ \delta (u)^2 ]
 & = & E\left[
  \left(
  \sum_{i=1}^n
  F_i \int_{B(R)} g_i(x) ( \gamma ( dx ) - \lambda ( dx ) )
  \right)^2
  \right]
  \\
& = &
E\left[
  \sum_{i,j=1}^n
  F_i F_j \int_{B(R)} g_i(x) ( \gamma ( dx ) - \lambda ( dx ) )
  \int_{B(R)} g_j(x) ( \gamma ( dx ) - \lambda ( dx ) )
  \right]
\\
& = &
2 E\left[
  \sum_{1\leq i<j\leq n}
  F_i \int_{B(R)} g_i(x) ( \gamma ( dx ) - \lambda ( dx ) )
  F_j \int_{B(R)} g_j(x) ( \gamma ( dx ) - \lambda ( dx ) )
  \right]
\\
 & & +
E\left[
  \sum_{i=1}^n
  F_i^2 \left( \int_{B(R)} g_i(x) ( \gamma ( dx ) - \lambda ( dx ) )
  \right)^2
  \right]
\\
& = &
E\left[
  \sum_{i=1}^n
  F_i^2 \int_{B(R)} g_i^2 (x) \lambda ( dx )
  \right]
\\
& = &
E\left[
\int_{B(R)} u^2 (x) \ \! \lambda ( dx )
  \right],
\end{eqnarray*}
which shows that \eqref{rel}
extends to the closure of ${\cal P}_0$ in $L^2(\Omega \times B(R))$
by density and a Cauchy sequence argument.
\end{Proofy}

\footnotesize

\def\cprime{$'$} \def\polhk#1{\setbox0=\hbox{#1}{\ooalign{\hidewidth
  \lower1.5ex\hbox{`}\hidewidth\crcr\unhbox0}}}
  \def\polhk#1{\setbox0=\hbox{#1}{\ooalign{\hidewidth
  \lower1.5ex\hbox{`}\hidewidth\crcr\unhbox0}}} \def\cprime{$'$}


TITLE: Unclear definition concerning convergence of a power series
QUESTION [1 upvotes]: I am following a course on computer algebra and at the end of the course, my professor wrote something down I could not follow at first. Concretely it handles about a definition given as follows:

Let $(b_n(x))_{n \geq 0}$ be a sequence of power series $\in \mathbb{K}[[x]]$.
Set $a_N := \sum_{n=0}^N b_n(x) \in \mathbb{K}[[x]]$ with $N \in \mathbb{N}$.
If $(a_N)_{N \geq 0}$ is convergent, i.e. $\exists b(x) \in \mathbb{K}[[x]]$ such that $\lim_\limits{N \to \infty} a_N(x) = b(x) = \lim_\limits{N \to \infty} \sum_{n=0}^N b_n(x) = \lim_\limits{n \to \infty} a_N(x) = b(x)$,
then we define $\sum_{n=0}^\infty b_n := b(x)$ $(= \lim_\limits{N \to \infty} \sum_{n = 0}^N b_n)$

Note that $\mathbb{K}[[x]]$ represents the ring of power series with Cauchy product and $\lim_\limits{n \to \infty} a_n(x)$ was introduced as notation for a convergent power series $a_n(x)$.
I tried to understand what the goal of this definition should be, but it is written down in such a confusing and sloppy way that I do not seem to get the sense of it anymore. I know we are working towards composition of formal power series, but I don't see why.
Because I need to solve some homeworks from this definition, I was wondering whether any one could guide me to references concerning this definition (e.g. does it have a name?) or explain what the goal of this definition could be.

REPLY [1 votes]: The definition may become easier to follow if we instead consider a comparable, more familiar situation.

Let's consider a real sequence
  $(b_n)_{n\geq 0}$ and the series
  \begin{align*}
\sum_{n=0}^\infty b_n\tag{1}
\end{align*}
  The series is defined in two ways. At one hand it is defined as sequence, namely  the sequence of it's partial sums
  \begin{align*}
\sum_{n=0}^\infty b_n:=\left(\sum_{n=0}^Nb_n\right)_{N\geq 0}\tag{2}
\end{align*}
  on the other hand it is defined as value. If the limit of the sequence of partial sums converges to a real value, we say the series (1) converges and define it's value as
  \begin{align*}
\sum_{n=0}^{\infty} b_n:= \lim_{N\rightarrow \infty}\left(\sum_{n=0}^Nb_n\right)\tag{3}
\end{align*}

You could find here a somewhat more detailed answer regarding this aspect.

In the framework of formal power series we can find similar definitions. We consider instead of real sequences a sequence of elements from $\mathbb{K}[[x]]$ which is $(b_n(x))_{n\geq 0}$ and want to define the series
  \begin{align*}
\sum_{n=0}^\infty b_n(x)\tag{4}
\end{align*}
  We do so on the one hand by considering the sequence of partial sums of the corresponding formal power series analogously  as we did in (2)
  \begin{align*}
\sum_{n=0}^\infty b_n(x):=\left(\sum_{n=0}^Nb_n(x)\right)_{N\geq 0}
\end{align*}
  This sequence of partial sums of formal power series may converge to an element $b(x)$ from $\mathbb{K}[[x]]$. In this case we define similarly to (3) the expression (4) as limit of the sequence of partial sums of formal power series.
  \begin{align*}
\sum_{n=0}^{\infty} b_n(x):= \lim_{N\rightarrow \infty}\left(\sum_{n=0}^Nb_n(x)\right)=b(x)
\end{align*}

Note that in the professor's text for convenience only $a_N(x)$ is used as shorthand to denote the partial sums
\begin{align*}
a_N(x):=\sum_{n=0}^Nb_n(x)\qquad\qquad N\geq 0
\end{align*}