Stein's method and normal approximation of Poisson functionals

Introduction

In a recent series of papers, Nourdin and Peccati , and Nourdin, Peccati and Réveillac , have shown that one can effectively combine Malliavin calculus on a Gaussian space (see e.g. ) and Stein’s method (see e.g. ) in order to obtain explicit bounds for the normal and non-normal approximation of smooth functionals of Gaussian fields.

The aim of the present paper is to extend the analysis initiated in to the framework of the normal approximation (in the Wasserstein distance) of regular functionals of Poisson measures defined on abstract Borel spaces. As in the Gaussian case, the main ingredients of our analysis are the following:

A set of Stein differential equations, relating the normal approximation in the Wasserstein distance to first order differential operators.

A (Hilbert space-valued) derivative operator $D$, acting on real-valued square-integrable random variables.

An integration by parts formula, involving the adjoint operator of $D$.

A “pathwise representation” of $D$ which, in the Poisson case, involves standard difference operators.

As a by-product of our analysis, we obtain substantial generalizations of the Central Limit Theorems (CLTs) for functionals of Poisson measures (for instance, for sequences of single and double Wiener-Itô integrals) recently proved in (see also for applications of these results to Bayesian non-parametric statistics). In particular, one of the main results of the present paper (see Theorem 5.1 below) is a CLT for sequences of multiple Wiener-Itô integrals of arbitrary (fixed) order with respect to a general Poisson measure. Our conditions are expressed in terms of contraction operators, and can be seen as a Poisson counterpart to the CLTs on Wiener space proved by Nualart and Peccati and Nualart and Ortiz-Latorre . The reader is referred to Decreusefond and Savy for other applications of Stein-type techniques and Malliavin operators to the assessment of Rubinstein distances on configuration spaces.

The remainder of the paper is organized as follows. In Section 2, we discuss some preliminaries, involving multiplication formulae, Malliavin operators, limit theorems and Stein’s method. In Section 3, we derive a general inequality concerning the Gaussian approximation of regular functionals of Poisson measures. Section 4 is devoted to upper bounds for the Wasserstein distance, and Section 5 to CLTs for multiple Wiener-Itô integrals of arbitrary order. Section 6 deals with sums of a single and a double integral. In Section 7, we apply our results to non-linear functionals of Ornstein-Uhlenbeck L vy processes.

Preliminaries

where $\mathfrak{P}\left(B\right)$ is a Poisson random variable with parameter $\mu\left(B\right)$. Note that properties (i)-(ii) imply, in particular, that $\widehat{N}$ is an independently scattered (or completely random) measure (see e.g. ).

where $\delta_{z}$ denotes the Dirac mass at $z$, and $\widehat{N}$ is the compensated canonical mapping given by

For every deterministic function $h\in L^{2}\left(Z,\mathcal{Z},\mu\right)=L^{2}\left(\mu\right)$ we write $\widehat{N}\left(h\right)=\int_{Z}h\left(z\right)\widehat{N}\left(dz\right)$ to indicate the Wiener-Itô integral of $h$ with respect to $\widehat{N}$ (see e.g. or ). We recall that, for every $h\in L^{2}\left(\mu\right)$, the random variable $\widehat{N}\left(h\right)$ has an infinitely divisible law, with Lévy-Khinchine exponent (see again ) given by

Fix $n\geqslant 2.$ We denote by $L^{2}\left(\mu^{n}\right)$ the space of real valued functions on $Z^{n}$ that are square-integrable with respect to $\mu^{n}$, and we write $L_{s}^{2}\left(\mu^{n}\right)$ to indicate the subspace of $L^{2}\left(\mu^{n}\right)$ composed of symmetric functions. For every $f\in L_{s}^{2}(\mu^{n})$, we denote by $I_{n}(f)$ the multiple Wiener-Itô integral of order $n$, of $f$ with respect to $\widehat{N}$. Observe that, for every $m,n\geqslant 2$, $f\in L_{s}^{2}(\mu^{n})$ and $g\in L_{s}^{2}(\mu^{m})$, one has the isometric formula (see e.g. ):

where the series converges in $L^{2}$ and, for each $n\geqslant 1$, the kernel $f_{n}$ is an element of $L_{s}^{2}(\mu^{n})$.

2 Contractions, stars and products

We now recall a useful version of the multiplication formula for multiple Poisson integrals. To this end, we define, for $q,p\geqslant 1$, $f\in L_{s}^{2}\left(\mu^{p}\right)$, $g\in L_{s}^{2}\left(\mu^{q}\right)$, $r=0,...,q\wedge p$ and $l=1,...,r$, the (contraction) kernel on $Z^{p+q-r-l}$, which reduces the number of variables in the product $fg$ from $p+q$ to $p+q-r-l$ as follows: $r$ variables are identified and, among these, $l$ are integrated out. This contraction kernel is formally defined as follows:

so that $f\star_{0}^{0}g(t_{1},\ldots,t_{p},s_{1},\ldots,s_{q})=f(t_{1},\ldots,t_{p})g(s_{1},\ldots,s_{q})$. For example, if $p=q=2$,

The quite suggestive ‘star-type’ notation is standard, and has been first used by Kabanov in (but see also Surgailis ). Plainly, for some choice of $f,g,r,l$ the contraction $f\star^{l}_{r}g$ may not exist, in the sense that its definition involves integrals that are not well-defined. On the positive side, the contractions of the following three types are well-defined (although possibly infinite) for every $q\geqslant 2$ and every kernel $f\in L_{s}^{2}(\mu^{q})$:

$f\star_{r}^{0}f(z_{1},....,z_{2q-r})$, where $r=0,....,q$, as obtained from (2.9), by setting $g=f$;

$f\star_{q}^{l}f(z_{1},...,z_{q-l})=\int_{Z^{l}}f^{2}(z_{1},...,z_{q-l},\cdot)d\mu^{l}$, for every $l=1,...,q$;

In particular, a contraction of the type $f\star_{q}^{l}f$, where $l=1,...,q-1$ may equal $+\infty$ at some point $(z_{1},...,z_{q-l})$. The following (elementary) statement ensures that any kernel of the type $f\star_{r}^{r}g$ is square-integrable.

Let $p,q\geqslant 1$, and let $f\in L_{s}^{2}(\mu^{q})$ and $g\in L_{s}^{2}(\mu^{p})$. Fix $r=0,...,q\wedge p$. Then, $f\star_{r}^{r}g\in L^{2}(\mu^{p+q-2r})$.

Proof. Just use equation (2.2) in the case $l=r$, and deduce the conclusion by a standard use of the Cauchy-Schwarz inequality.

We also record the following identity (which is easily verified by a standard Fubini argument), valid for every $q\geqslant 1$, every $p=1,...,q$:

for every $f\in L_{s}^{2}(\mu^{q})$. The forthcoming product formula for two Poisson multiple integrals is proved e.g. in and .

Let $f\in L_{s}^{2}\left(\mu^{p}\right)$ and $g\in L_{s}^{2}\left(\mu^{q}\right)$, $p,q\geqslant 1$, and suppose moreover that $f\star_{r}^{l}g\in L^{2}(\mu^{p+q-r-l})$ for every $r=0,...,p\wedge q$ and $l=1,...,r$ such that $l\neq r$. Then,

where the tilde “ $\widetilde{}$ ” stands for symmetrization, that is,

where $\sigma$ runs over all $\left(q+p-r-l\right)!$ permutations of the set $\left\{1,...,q+p-r-l\right\}$.

where: (i) $\{W_{t},\,t\geq 0\}$ is a standard Brownian motion, (ii)

(with convergence in $L^{2}(\Omega)$), that is, $M$ is a centered independently scattered random measure such that

3 Four Malliavin-type operators

In what follows, given $f\in L^{2}_{s}(\mu^{q})$ ($q\geqslant 2$) and $z\in Z$, we write $f(z,\cdot)$ to indicate the function on $Z^{q-1}$ given by $(z_{1},...,z_{q-1})\rightarrow f(z,z_{1},...,z_{q-1})$.

If $F$ verifies (2.16) (that is, if $F\in{\rm dom}D$), then the random function $z\rightarrow D_{z}F$ is given by

For instance, if $F=I_{1}(f)$, then $D_{z}F$ is the non-random function $z\rightarrow f(z)$. If $F=I_{2}(f)$, then $D_{z}F$ is the random function

A version of the following result, whose content is one of the staples of our analysis, is proved e.g. in [19, Theorem 6.2] (see also [28, Proposition 5.1]). One should also note that the proof given in applies to the framework of a Radon control measure $\mu$: however, the arguments extend immediately to the case of a Borel control measure considered in the present paper, and we do not reproduce them here (see, however, the remarks below). It relates $F_{z}$ to $D_{z}F$ and provides a representation of $D$ as a difference operator.

In the Itô-type framework detailed in Remark 2.6, we could alternatively use the definition of the Malliavin derivative introduced in , that is,

When applied to the case $F=I_{1}(f)$, formula (2.20) is just a consequence of the straightforward relation

To have an intuition of the proof of (2.20) in the general case, consider for instance a random variable of the type $F=\widehat{N}(A)\times\widehat{N}(B)$, where the sets $A,B\in\mathcal{Z}_{\mu}$ are disjoint. Then, one has that $F=I_{2}(f)$, where $f(x,y)=2^{-1}\{{\bf 1}_{A}(x){\bf 1}_{B}(y)+{\bf 1}_{A}(y){\bf 1}_{B}(x)\}$. Using (2.18), we have that

Now fix $z\in Z$. There are three possible cases: (i) $z\notin A$ and $z\notin B$, (ii) $z\in A$, and (iii) $z\in B$. If $z$ is as in (i), then $F_{z}=F$. If $z$ is as in (ii) (resp. as in (iii)), then

As a consequence, one deduces that $F_{z}-F=\widehat{N}(B){\bf 1}_{A}(z)+\widehat{N}(A){\bf 1}_{B}(z)$, so that relation (2.20) is obtained from (2.21).

Observe also that Lemma 2.7 yields that, if $F,G\in{\rm dom}D$ are such that $FG\in{\rm dom}D$, then

(see [19, Lemma 6.1] for a detailed proof of this fact).

If $u\in{\rm dom}\delta$, then the random variable $\delta(u)$ is defined as

For every $G\in{\rm dom}D$ and every $u\in{\rm dom}\delta$, one has that

If $F\in{\rm dom}L$, then the random variable $LF$ is given by

For every $F\in{\rm dom}L$, one has that $F\in{\rm dom}D$ and $DF\in{\rm dom}\delta$. Moreover,

Proof. The first part of the statement is easily proved by applying the definitions of ${\rm dom}D$ and ${\rm dom}\delta$ given above. In view of Proposition 2.3, it is now enough to prove (2.28) for a random variable of the type $F=I_{q}(f)$, $q\geqslant 1$. In this case, one has immediately that $D_{z}F=qI_{q-1}(f(z,\cdot))$, so that $\delta DF=qI_{q}(f)=-LF$.

4 Normal approximation in the Wasserstein distance, via Stein’s method

Let $W$ be a random variable. Then, $W\stackrel{{\scriptstyle\rm Law}}{{=}}X\sim\mathscr{N}(0,1)$ if, and only if,

If $h$ is absolutely continuous with bounded derivative, then (2.31) has a solution $f_{h}$ which is twice differentiable and such that $\|f_{h}^{\prime}\|_{\infty}\leqslant\|h^{\prime}\|_{\infty}$ and $\|f_{h}^{\prime\prime}\|_{\infty}\leqslant 2\|h^{\prime}\|_{\infty}$.

Let $\mathscr{F}_{W}$ denote the class of twice differentiable functions, whose first derivative is bounded by 1 and whose second derivative is bounded by 2. If $h\in{\rm Lip}(1)$ and thus $\|h^{\prime}\|_{\infty}\leqslant 1$, then the solution $f_{h}$, appearing in Lemma 2.11(ii), satisfies $\|f_{h}^{\prime}\|_{\infty}\leqslant 1$ and $\|f_{h}^{\prime\prime}\|_{\infty}\leqslant 2$, and hence $f_{h}\in\mathscr{F}_{W}$.

A general inequality involving Poisson functionals

The following estimate, involving the normal approximation of smooth functionals of $\widehat{N}$, will be crucial for the rest of the paper. We use the notation introduced in the previous section.

Moreover, if $F$ has the form $F=I_{q}(f)$, where $q\geqslant 1$ and $f\in L^{2}_{s}(\mu^{q})$, then

By the fact that $\|f^{\prime}\|_{\infty}\leqslant 1$ and by Cauchy-Schwarz,

On the other hand, one sees immediately that

Relations (3.3) and (3.4) are immediate consequences of the definition of $L^{-1}$ given in (2.29). The proof is complete.

Note that, in general, the bounds in formulae (3.1)–(3.2) can be infinite. In the forthcoming Sections 4–7, we will exhibit several examples of random variables in ${\rm dom}D$ such that the bounds in the statement of Theorem 3.1 are finite.

Let $G$ be an isonormal Gaussian process (see ) over some separable Hilbert space $\EuFrak{H}$, and suppose that $F\in L^{2}(\sigma(G))$ is centered and differentiable in the sense of Malliavin. Then, the Malliavin derivative of $F$, noted $DF$, is a $\EuFrak{H}$-valued random element, and in it is proved that one has the following upper bound on the Wasserstein distance between the law of $F$ and the law of $X\sim\mathscr{N}(0,1)$:

where $L^{-1}$ is the inverse of the Ornstein-Uhlenbeck generator associated with $G$.

Now fix $h\in L^{2}(\mu)$. It is clear that the random variable $I_{1}(h)=\widehat{N}(h)$ is an element of ${\rm dom}D$. One has that $D_{z}\widehat{N}(h)=h(z)$; moreover $-L^{-1}\widehat{N}(h)=\widehat{N}(h)$. Thanks to these relations, one deduces immediately from Theorem 3.1 the following refinement of Part A of Theorem 3 in .

Let $h\in L^{2}(\mu)$ and let $X\sim\mathscr{N}(0,1)$. Then, the following bound holds:

As a consequence, if $\mu(Z)=\infty$ and if a sequence $\{h_{k}\}\subset L^{2}(\mu)\cap L^{3}(\mu)$ verifies, as $k\rightarrow\infty$,

and the inequality (3.6) provides an explicit upper bound in the Wasserstein distance.

In the following section, we will use Theorem 3.1 in order to prove general bounds for the normal approximation of multiple integrals of arbitrary order. As a preparation, we now present two simple applications of Corollary 3.4.

which is consistent with the usual Berry-Ess en estimates.

Fix $\lambda>0$. We consider the Ornstein-Uhlenbeck Lévy process given by

We shall show that Corollary 3.4 implies the existence of a finite constant $q_{\lambda}>0$, depending uniquely on $\lambda$, such that, for every $T>0$

To see this, first use a Fubini theorem in order to represent $A_{T}^{\lambda}$ as an integral with respect to $\widehat{N}$, that is, write

Clearly, $A_{T}^{\lambda}\in{\rm dom}D$ and $DA_{T}^{\lambda}=h_{T}^{\lambda}$. By using the computations contained in [23, Proof of Theorem 4], one sees immediately that there exists a constant $\beta_{\lambda}$, depending uniquely on $\lambda$ and such that

Since one has also that $|\|h_{T}^{\lambda}\|^{2}_{L^{2}(\mu)}-1|=O(1/T)$, the estimate (3.10) is immediately deduced from Corollary 3.4.

Bounds on the Wasserstein distance for multiple integrals of arbitrary order

In this section we establish general upper bounds for multiple Wiener-It integrals of arbitrary order $q\geqslant 2$, with respect to the compensated Poisson measure $\widehat{N}$. Our techniques hinge on the forthcoming Theorem 4.2, which uses the product formula (2.14) and the inequality (3.2).

Fix $q\geqslant 2$ and let $f\in L^{2}_{s}(\mu^{q})$. The operator $G_{p}^{q}$ transforms the function $f$, of $q$ variables, into a function $G_{p}^{q}f$ of $p$ variables, where $p$ can be as large as $2q$. When $p=0$, we set

where the ‘star’ contractions have been defined in formulae (2.2) and (2.9), and the tilde “$\,\widetilde{}\,$ ” indicates a symmetrization. The notation (4.1) is mainly introduced in order to give a more compact representation of the RHS of (2.14) when $g=f$. Indeed, suppose that $f\in L^{2}_{s}(\mu^{q})$ ($q\geqslant 2$), and that $f\star_{r}^{l}f\in L^{2}(\mu^{2q-r-l})$ for every $r=0,...,q$ and $l=1,...,r$ such that $l\neq r$; then, by using (2.14) and (4.1), one deduces that

where $I_{0}(G_{0}^{q}f)=G_{0}^{q}f=q!\|f\|^{2}_{L^{2}(\mu^{q})}$. Note that the advantage of (4.2) (over (2.14)) is that the square $I_{q}(f)^{2}$ is now represented as an orthogonal sum of multiple integrals. As before, given $f\in L^{2}_{s}(\mu^{q})$ ($q\geqslant 2$) and $z\in Z$, we write $f(z,\cdot)$ to indicate the function on $Z^{q-1}$ given by $(z_{1},...,z_{q-1})\rightarrow f(z,z_{1},...,z_{q-1})$. To simplify the presentation of the forthcoming results, we now introduce a further assumption.

Assumption A. For the rest of the paper, whenever considering a kernel $f\in L_{s}^{2}(\mu^{q})$, we will implicitly assume that every contraction of the type

is well-defined and finite for every $r=1,...,q$, every $l=1,...,r$ and every $(z_{1},...,z_{2q-r-l})\in Z^{2q-r-l}$.

Assumption A ensures, in particular, that the following relations are true for every $r=0,....,q-1$, every $l=0,...,r$ and every $(z_{1},...,z_{2(q-1)-r-l})\in Z^{2(q-1)-r-l}$:

(note that the symmetrization on the LHS of the second equality does not involve the variable $z$).

The notation $G_{p}^{q-1}f(z,\cdot)$ ($p=0,...,2(q-1)$) stands for the action of the operator $G_{p}^{q-1}$ (defined according to (4.1)) on $f(z,\cdot)$, that is,

For instance, if $f\in L^{2}_{s}(\mu^{3})$, then

Note also that, for a fixed $z\in Z$, the quantity $G_{0}^{q-1}f(z,\cdot)$ is given by the following constant

Finally, for every $q\geqslant 2$ and every $f\in L^{2}_{s}(\mu^{q})$, we set

where in (4.7) we have used the change of variables $t=r+1$ and $s=l+1$, as well as the fact that $p\geqslant 1$. We stress that the symmetrization in (4.6) does not involve the variable $z$.

2 Bounds on the Wasserstein distance

When $q\geqslant 2$ and $\mu(Z)=\infty$, we shall focus on kernels $f\in L^{2}_{s}(\mu^{q})$ verifying the following technical condition: for every $p=1,...,2(q-1)$,

As will become clear from the subsequent discussion, the requirement (4.8) is used to justify a Fubini argument.

When $q=2$, one deduces from (4.3) that $G_{1}^{1}f(z,\cdot)(x)=f(z,x)^{2}$ and $G_{2}^{1}f(z,\cdot)(x,y)=f(z,x)f(z,y)$. It follows that, in this case, condition (4.8) is verified if, and only if, the following relation holds:

Indeed, since $f$ is square-integrable, the additional relation

From relation (4.3), one deduces immediately that (4.8) is implied by the following (stronger) condition: for every $p=1,...,2(q-1)$ and every $(r,l)$ such that $2(q-1)-r-l=p$

condition (4.8) is automatically satisfied (to see this, just apply the Cauchy-Schwarz inequality).

Arguing as in the previous point, a sufficient condition for (4.8) to be satisfied, is that the support of the symmetric function $f$ is contained in a set of the type $A\times\cdot\cdot\cdot\times A$ where $A$ is such that $\mu(A)<\infty$ .

Fix $q\geqslant 2$ and let $X\sim\mathscr{N}(0,1)$. Let $f\in L^{2}_{s}(\mu^{q})$ be such that:

whenever $\mu(Z)=\infty$, condition (4.8) is satisfied for every $p=1,...,2(q-1)$;

for $d\mu$-almost every $z\in Z$, every $r=1,...,q-1$ and every $l=0,...,r-1$, the kernel $f(z,\cdot)\star_{r}^{l}f(z,\cdot)$ is an element of $L^{2}_{s}(\mu^{2(q-1)-r-l})$.

Denote by $I_{q}(f)$ the multiple Wiener-It integral, of order $q$, of $f$ with respect to $\widehat{N}$. Then, the following bound holds:

where the notations $\widehat{G}^{q}_{p}f$ and $G_{p}^{q-1}f(z,\cdot)$ are defined, respectively, in (4.5)–(4.7) and (4.3). Moreover, the bound appearing on the RHS of (4.12)–(4.13) can be assessed by means of the following estimate:

There are contraction norms in (4.18) that do not appear in the previous formula (4.15), and vice versa. For example, in (4.18) one has $\|f\star_{b}^{0}f\|_{L^{2}(\mu^{2q-b})}$, where $b=1,...,q$ (this corresponds to the case $b\in\{1,...,q\}$ and $a=0$). By using formula (2.13), these norms can be computed as follows:

We stress that $f^{2}=f\star_{q}^{0}f$, and therefore $f\star_{q}^{0}f\in L^{2}(\mu^{q})$ if, and only if, $f\in L^{4}(\mu^{q})$.

One should compare Theorem 4.2 with Proposition 3.2 in , which provides upper bounds for the normal approximation of multiple integrals with respect to an isonormal Gaussian process. The bounds in are also expressed in terms of contractions of the underlying kernel.

To obtain (4.21), observe first that, since assumption (a) above is in order, then relations (4.12)–(4.13) in the statement of Theorem 4.2 yield that

since $\|f\star_{2}^{1}f\|_{L^{2}(\mu)}=\|f\star_{1}^{0}f\|_{L^{2}(\mu^{3})}$. A general statement, involving random variables of the type $F=I_{1}(g)+I_{2}(h)$ is given in Theorem 6.1.

Proof of Theorem 4.2. First observe that, according to Theorem 3.1, we have that

The rest of the proof is divided in four steps:

Proof of the estimate displayed in formulae (4.14)–(4.15).

Proof of the estimate in formulae (4.17)–(4.18), under the assumption (4.16).

Step (S1). Start by defining $A_{f}$ as the collection of those $z\in Z$ such that Assumption (ii) in the statement is violated, that is,

By assumption, one has that $\mu(A_{f})=0$. By writing

For every $z\in Z$, every $r=1,...,q-1$ and every $l=0,...,r-1$, one has that $f(z,\cdot)\star_{r}^{l}f(z,\cdot)\in L^{2}_{s}(\mu^{2(q-1)-r-l})$.

Now use (2.17) to write $D_{z}I_{q}(f)=qI_{q-1}(f(z,\cdot))$, $z\in Z$. By virtue of the multiplication formula (2.14), in particular (4.2), and by adopting the notation (4.1), we infer that, for every $z\in Z$

Since (4.8) is in order, one has that, for every $p=1,...,2(q-1)$,

where we have used the Cauchy-Schwarz inequality, combined with the isometric properties of multiple integrals, as well as the fact that, for every $z\in Z$,

Relations (4.23)–(4.24) yield that one can write

Since assumption (4.8) is in order, one has that, for every $p=1,...,2(q-1)$,

and one can easily verify that, for $1\leqslant p\neq l\leqslant 2(q-1)$, the random variables

The proof of (4.27) can be achieved by using the following relations:

Note that the use of the Fubini theorem in the equality (4.28) is justified by the chain of inequalities (4.26), which is in turn a consequence of assumption (4.8).

Now use the Cauchy-Schwarz inequality, in order to write

By using (4.4) and (4.23), one deduces immediately that

Step (S3). By using several times the inequality $\sqrt{a+b}\leqslant\sqrt{a}+\sqrt{b}$ ($a,b\geqslant 0$) one sees that, in order to prove (4.14)–(4.15), it is sufficient to show that

To see this, use (4.7) and the fact that (by (2.6)) $\|\widetilde{f\star_{t}^{s}f}\|_{L^{2}(\mu^{2q-t-s})}\leqslant\|f\star_{t}^{s}f\|_{L^{2}(\mu^{2q-t-s})}$, to obtain that

Step (S4). By using (4.3) and some standard estimates, we deduce that

We claim that, if (4.16) is satisfied, then, for every $r=0,...,q-1$ and $l=0,...,r$

relation (4.34) can be deduced by a standard use of the Fubini theorem (assumption (4.16) is not needed here). Now fix $p=1,...,2q-2$, as well as $r=1,...q-2$ and $l=0,...,r$ in such a way that $2(q-1)-r-l=p$. For every fixed $z\in Z$, write $|f(z,\cdot)|\star^{l}_{r}|f(z,\cdot)|$ to indicate the contraction of indices $(r,l)$ obtained from the positive kernel $|f(z,\cdot)|$. Note that, for $z$ fixed, such a contraction is a function on $Z^{p}$, and also, in general, $|f(z,\cdot)|\star^{l}_{r}|f(z,\cdot)|\geqslant f(z,\cdot)\star^{l}_{r}f(z,\cdot)$ and $|f(z,\cdot)|\star^{l}_{r}|f(z,\cdot)|\neq f(z,\cdot)\star^{l}_{r}f(z,\cdot)$. By a standard use of the Cauchy-Schwarz inequality and of the Fubini theorem, one sees that

where the last relation is a consequence of assumption (4.16) as well as of the fact that, by construction, $1\leqslant 1+r-l\leqslant q-1$. Relation (4.35) implies that one can apply the Fubini Theorem to the quantity

(by first writing the contractions $f(z,\cdot)\star^{l}_{r}f(z,\cdot)$ in an explicit form), so to obtain the desired equality (4.34). By plugging (4.34) into (4.32) and by applying the change of variables $a=q-1-r$ and $b=q-l$, one deduces (4.17)–(4.18). This concludes the proof of Theorem 4.2.

Central limit theorems

We consider here CLTs. The results of this section generalize the main findings of . The following result uses Theorem 4.2 in order to establish a general CLT for multiple integrals of arbitrary order.

For every $k\geqslant 1$, the kernel $f_{k}$ verifies (4.8) for every $p=1,...,2(q-1)$.

For every $r=1,...,q$, and every $l=1,...,r\!\wedge\!(q-1)$, one has that $f_{k}\star_{r}^{l}f_{k}\!\in\!L^{2}(\mu^{2q-r-l})$ and also $\|f_{k}\star_{r}^{l}f_{k}\|_{L^{2}(\mu^{2q-r-l})}\rightarrow 0$ (as $k\rightarrow\infty$).

For every $k\geqslant 1$, one has that $\int_{Z^{q}}f^{4}_{k}\,\,d\mu^{q}<\infty$ and, as $k\rightarrow\infty$, $\int_{Z^{q}}f^{4}_{k}\,\,d\mu^{q}\rightarrow 0$.

Then, $F_{k}\stackrel{{\scriptstyle\rm law}}{{\rightarrow}}X$, as $k\rightarrow\infty$, and formulae (4.14)–(4.20) provide explicit bounds in the Wasserstein distance $d_{W}(I_{q}(f_{k}),X)$.

This last fact coincides with the content of Part 1 of Theorem 2 in , where one can find an alternate proof based on a decoupling technique, known as the “principle of conditioning” (see e.g. Xue ). Note that an explicit upper bound for the Wasserstein distance can be deduced from relation (4.21).

Consider a sequence of random variables of the type $I_{3}(f_{k})$, $k\geqslant 1$, with unitary variance and verifying Assumption (I) in the statement of Theorem 5.1. Then, according to the conclusion of Theorem 5.1, a sufficient condition in order to have that, as $k\rightarrow\infty$,

is that the following six quantities converge to zero:

Moreover, an explicit upper bound in the Wasserstein distance is given by the estimate (4.5).

The following result, proved in [23, Theorem 2], represents a counterpart to the CLTs for double integrals discussed in Example 5.2.

Consider a sequence $F_{k}=I_{2}(f_{k})$, $k\geqslant 1$, of double integrals verifying assumptions (a), (b) and (c) of Example 5.2. Suppose moreover that (5.2) takes place. Then,

if the sequence $\left\{F_{k}^{4}:k\geqslant 1\right\}$ is uniformly integrable, then conditions (5.1), (5.3) and (5.4) are equivalent.

For the time being, it seems quite hard to prove a result analogous to Proposition 5.4 for a sequence of multiple integrals of order $q\geqslant 3$.

Sum of a single and a double integral

As we will see in the forthcoming Section 7, when dealing with quadratic functionals of stochastic processes built from completely random measures, one needs explicit bounds for random variables of the type $F=I_{1}(g)+I_{2}(h)$, that is, random variables that are the sum of a single and a double integral. The following result, that can be seen as a generalization of Part B of Theorem 3 in , provides explicit bounds for random variables of this type.

The function $g$ belongs to $L^{2}(\mu)\cap L^{3}(\mu)$;

The kernel $h\in L^{2}_{s}(\mu^{2})$ is such that: (a) $h\star_{2}^{1}h\in L^{2}(\mu^{1})$, (b) relation (4.9) is verified, with $h$ replacing $f$, and (c) $h\in L^{4}(\mu^{2})$.

Then, one has the following upper bound on the Wasserstein distance between the law of $F$ and the law of $X\sim\mathscr{N}(0,1)$:

Proof. Thanks to Theorem 3.1, we know that $d_{W}(F,X)$ is less or equal to the RHS of (3.2). We also know that

By using the multiplication formula (2.14) (in the case $p=q=1$) as well as a Fubini argument, one easily deduces that

To conclude the proof, one shall use the following relations, holding for every real $a,b$:

By applying (6.4) in the case $a=g(z)$ and $b=I_{1}(h(z,\cdot))$, one deduces that

By using the Cauchy-Schwarz inequality, one infers that

Formula (6.2) is once again an elementary consequence of the Cauchy-Schwarz inequality.

Consider a sequence of vectors $(F_{k},H_{k})$, $k\geqslant 1$ such that: (i) $F_{k}=I_{2}(f_{k})$, $k\geqslant 1$, is a sequence of double integrals verifying assumptions (a), (b) and (c) in Example 5.2, and (ii) $H_{k}=I_{1}(h_{k})$, $k\geqslant 1$, is a sequence of single integrals with unitary variance. Suppose moreover that the asymptotic relations in (3.7), (5.2) and (5.3) take place. Then, the estimates (6.1)–(6.2) yield that, for every $(\alpha,\beta)\neq(0,0)$, the Wasserstein distance between the law of

and the law of $X\sim\mathscr{N}(0,1)$, converges to zero as $k\rightarrow\infty$, thus implying that $(F_{k},H_{k})$ converges in law to a vector $(X,X^{\prime})$ of i.i.d. standard Gaussian random variables. Roughly speaking, this last fact implies that, when assessing the asymptotic joint Gaussianity of a vector such as $(F_{k},H_{k})$, one can study separately the one-dimensional sequences $\{F_{k}\}$ and $\{H_{k}\}$. This phenomenon coincides with the content of Part B of Theorem 3 in . See , and for similar results involving vectors of multiple integrals (of arbitrary order) with respect to Gaussian random measures.

Applications to non-linear functionals of Ornstein-Uhlenbeck L vy processes

As an illustration, in this section we focus on CLTs related to Ornstein-Uhlenbeck L vy processes, that is, processes obtained by integrating an exponential kernel of the type

with respect to an independently scattered random measure. Ornstein-Uhlenbeck L vy processes have been recently applied to a variety of frameworks, such as finance (where they are used to model stochastic volatility – see e.g. ) or non-parametric Bayesian survival analysis (where they represent random hazard rates – see e.g. ). In particular, in the references and it is shown that one can use some of the CLTs of this section in the context of Bayesian prior specification.

We consider the stationary Ornstein-Uhlenbeck Lévy process given by

The following result has been proved in [23, Theorem 5].

For every $\lambda>0$, as $T\rightarrow\infty$,

where $c^{2}_{\nu}:=\int u^{4}\nu(du)$ and $X\sim\mathscr{N}\left(0,1\right)$ is a centered standard Gaussian random variable.

By using Theorem 6.1, one can obtain the following Berry-Esséen estimate on the CLT appearing in (7.2) (compare also with Example 3.6).

Let $Q(T,\lambda)$, $T>0$, be defined as in (7.2), and set

Then, there exists a constant $0<\gamma(\lambda)<\infty$, independent of $T$ and such that

As a consequence of the multiplication formula (2.14) and of a standard Fubini argument, one has (see [23, Proof of Theorem 5])

a combination of a single and of a double integral. To apply Theorem 6.1, we use the following asymptotic relations (that one can verify by resorting to the explicit definitions of $H_{\lambda,T}$ and $H^{\ast}_{\lambda,T}$ given in (7.1)), holding for $T\rightarrow\infty$:

The conclusion is obtained by using the estimates (6.1)–(6.2) and applying Theorem 6.1.

2 Berry-Esséen bounds for arbitrary tensor powers of Ornstein-Uhlenbeck kernels

We sometimes set $y=(u,x)$. Note that, for every $t\geqslant 0$, one has that $\int f^{2}_{t}(y)\mu(dy)=\int f^{2}_{t}(u,x)\nu(du)dx=1$, and therefore $f_{t}^{\otimes q}\in L_{s}^{2}(\mu^{q})$; it follows that the multiple integral

is well defined for every $t\geqslant 0$.

Fix $q\geqslant 2$, and suppose that $\int|u|^{j}d\nu<\infty$, $\forall j=1,...,2q$. Then, one can prove that $Z_{t}(1)^{q}$ is square-integrable and also that the random variable $Z_{t}(q)$ coincides with the projection of $Z_{t}(1)^{q}$ on the $q$th Wiener chaos associated with $\widehat{N}$. This fact can be easily checked when $q=2$: indeed (using Proposition 2.5 in the case $f=g=f_{t}$) one has that

thus implying the desired relation. The general case can be proved by induction on $q$.

The main result of this section is the following application of Theorem 4.2 and Theorem 5.1.

Fix $\lambda>0$ and $q\geqslant 2$, and define the positive constant $c=c(q,\lambda):=2(q-1)!/\lambda$. Then, one has that, as $T\rightarrow\infty$,

and there exists a finite constant $\rho=\rho(\lambda,q,\nu)>0$ such that, for every $T>0$,

Proof. The crucial fact is that, for each $T$, the random variable $M_{T}(q)$ has the form of a multiple integral, that is, $M_{T}(q)=I_{q}(F_{T})$, where $F_{T}\in L_{s}^{2}(\mu^{q})$ is given by

(relation (7.9) improves the bounds in Theorem 4.2). In order to prove (7.9)–(7.11), for every $t_{1},t_{2}\geqslant 0$ we introduce the notation

(recall that $\int u^{2}d\nu=1$) and also, for $t_{1},t_{2},t_{3},t_{4}\geqslant 0$,

To prove (7.9), one uses the relation (7.12) to get

In the remaining of the proof, we will write $\kappa$ in order to indicate a strictly positive finite constant independent of $T$, that may change from line to line. To prove (7.10), one uses the fact that

where the last relation is obtained by resorting to the explicit representation (7.14), and then by evaluating the restriction of the quadruple integral to each simplex of the type $\{t_{\pi(1)}>t_{\pi(2)}>t_{\pi(3)}>t_{\pi(4)}\}$, where $\pi$ is a permutation of the set $\{1,2,3,4\}$. We shall now verify the class of asymptotic relations (7.11) for $r=q$ and $l=1,...,q$. With $y=(u,x)$, one has

where the last relation is verified by first using (7.12)–(7.14), and then by assessing the restriction of the quadruple integral to each one of the $4!=24$ simplexes of the type $\{t_{\pi(1)}>t_{\pi(2)}>t_{\pi(3)}>t_{\pi(4)}\}$. To deal with (7.11) in the case $r=1,...,q-1$ and $l=1,...,r$, one uses the fact that

where the last relation is once again obtained by separately evaluating each restriction of the quadruple integral over a given simplex. This concludes the proof.

Acknowledgments. Part of this paper has been written while G. Peccati was visiting the Department of Mathematics of the “Universitat Autónoma de Barcelona”, in April 2008. This author heartily thanks J.L. Solé and F. Utzet for their hospitality and support. J.L. Solé and F. Utzet were supported by the Grant BFM2006-06427, Ministerio de Educación y Ciencia and FEDER. M.S. Taqqu was partially supported by the NSF Grants DMS-0505747 and DMS-0706786 at Boston University.

References