## Malliavin-Stein Method for Muti-dimensional U-statistics of Poisson point processes

Here is my new manuscript submitted to ARXIV [math.PR]. The preprint version can be found here.

**MALLIAVIN-STEIN METHOD FOR MULTI-DIMENSIONAL**

**U-STATISTICS OF POISSON POINT PROCESSES**

**Abstract.** *In this paper, we give an upper bound for a probabilistic distance between a Gaussian vector and a vector of U-statistics of Poisson point processes by applying Malliavin-Stein inequality on the Poisson space.*

**I – Introduction**

The theory of Malliavin calculus on the Poisson space was firstly studied by Nualart and Vives in their excellent paper of Strasbourg’s seminars *[Nualart1990]*. For some important contributions and applications for Poisson point processes, we refer to *[Houdre1995, Wu2000, Peccati2011, Last2011]*. The combination of Stein’s method and Malliavin calculus on the Poisson space related to the normal approximations of Poisson functionals has been considered by Peccati, Solé, Taqqu, Utzet and Zheng in their recent papers *[Peccati2010a, Peccati2010b]*.

The basic theory of U-statistics was introduced by Hoeffding *[Hoeffding1948]* as a class of statistics that is especially important in estimation theory. Further applications are widely regarded in theory of random graphs, spatial statistics, theory of communication and stochastic geometry, see e.g. *[Lee1990, Koroljuk1994, Borovskikh1996]*.

Recently, the idea of central limit theorems for U-statistics of Poisson point processes by using Malliavin calculus and Stein’s method has been given by Reitzner and Schulte in *[Reitzner2011]*. Our main work in the current paper is to extend their results to vectors of U-statistics by applying the muti-dimensional Malliavin-Stein inequality, that was proved by Peccati and Zheng in *[Peccati2010b]*. Some preliminaries of Malliavin calculus on the Poisson space and U-statistic will be introduced in Section 2. An upper bound for a probabilistic distance between a Gaussian vector and a square integrable random variable with finite Wiener-Itô chaos expansions will be shown in Section 3 and its application for multi-dimensional U-statistics of Poisson point processes will be given in Section 4.

**II – Malliavin Calculus on the Poisson space**

Let be some measure space with -finite measure . The Poisson point process or Poisson measure with intensity measure is a family of random variables defined on some probability space such that

1. , is a Poisson random variable with rate .

2. If sets don’t intersect then the corresponding random variables from i) are mutually independent.

3. , is a measure on .

Note that, for some -finite measure space , we can always set

where denotes the Dirac mass at , and for each , we give the mapping such that

Moreover, the -field is supposed to be the -completion of the -field generated by .

Let denote the space of all integer-valued -finite measures on , which can be equipped with the smallest -algebra such that for each then the mapping is measurable. We can give on the probability measure induced by the Poisson measure and denote as the set of all measurable functions such that , where the expectation takes w.r.t. the probability measure .

Let be the space of all measurable functions such that

Note that becomes a Hilbert space when we define on it the scalar product

We denote as the subset of symmetric functions in , in the sense that the functions are invariant under all permutations of the arguments. Now, for each , we define the random variable , which is also known as a compensated Poisson measure.

For each symmetric function , one can define the multiple Wiener-It\^o integral w.r.t the compensated Poisson measure denoted by

At first, let be the class of simple functions , which takes the form

where and the sets are pairwise disjoint such that is symmetric and vanishes on diagonals, that means if for some . The multiple Wiener-It\^o integral for a simple function in the form with respect to the compensated Poisson measure is defined by

Since the class is dense in then for every , there exists a sequence such that in . Moreover, one can show that . Hence, the the multiple Wiener-It\^o integral for a symmetric function can be defined as the limit of the sequence in and we denote it as .

**Proposition 2.1.** *For and , , then
1. ,
2.
where is the Kronecker delta*.

For a measurable function and we define the difference operator as

where is the Dirac measure at the point . The iterated difference operator is given by

We define the kernels of as functions given by

Note that is a symmetric function.

We define the Ornstein-Uhlenbeck generator as

**Proposition 2.2.*** For each , then the kernels are elements of , and uniquely admit the Wiener-It\^o chaos expansion in the form
*

where the sum converges in . Furthermore, for

**Proposition 2.3.** *Let , and assume that
*

Then the difference of at is given by

**Proposition 2.4.*** For each random variable such that
*

then the Ornstein-Uhlenbeck generator is calculated as

Moreover, its inverse operator is calculated as

for each such that .

For more details of the Malliavin calculus on Poisson space, we refer the reader to *[Nualart1990]*.

**III – Multi-dimensional Malliavin-Stein inequality**

In the next sequence, we use the probabilistic distance of two -dimensional random vectors such that , which is defined by

where is the family of all real-valued functions such that

In the above inequality,

stands for the Hessian matrix of evaluated at a point and we use the notation of operator norm for a real matrix given by

**
Theorem 3.1.** [Multi-dimensional Malliavin-Stein inequality]

*Consider a random vector such that for , , and . Suppose that , where is a positive definite symmetric matrix. Then,*

For the proof, we refer to [Peccati2010b].

Now we consider a -dimensional random vector with the covariance matrix such that each component has finite Wiener-It\^o chaos expansions with kernels , which vanishes if . Let give the centered random vector

where stands for the square root of a positive definite matrix , i.e if has the eigenvalues decomposition

then

Let us use vector notations

and note that the inequality

holds for all positive definite matrices .

Therefore, by the properties matrix trace and using the inequality , we have

where

denotes the Frobenius norm of matrix .

Note that

and

Hence,

It follows that

Moreover, by using Holder inequality and the property of matrix norm, we have

Substituting and to the inequality in Theorem 3.1 for , we obtain that

**Theorem 3.2.** *Let give a -dimensional Gaussian random variable . Assume that such that and has finite Wiener-It\^o chaos expansions with kernels , which vanishes if . Then
*

**IV – Application for multi-dimensional U-statistics**

In this section we consider the -dimensional vector of U-statistics of the Poisson point process

where , and denotes the set of all -tuples of distinct points of . This means that each component

is an U-statistic of order with respect to the Poisson point process , .

The following properties of (one-dimensional) U-statistics are obtained by Reitzner and Schulte in *[Reitzner2011]*

**Proposition 4.1.** *Let be a U-statistic of order in the form
*

Then the kernels of the Wiener-It\^o chaos expansion of have the form

**Proposition 4.2.** *Assume , then
1. If is a U-statistic, then has a finite Wiener-It\^o chaos expansion with kernels , .
2. If has a finite Wiener-Itô chaos expansion with kernels , , then is a finite sum of U-statistics and a constant.*

**Proposition 4.3.** *Let and be the set of all partitions of such that for each ,
1. , are always in different subsets of , and such that
2. every subset of has at least two elements.
For every partition we define an operator that replaces all elements of in that belong to the same subset of by a new variable , , where denotes the number of subsets of the partition . Then
*

Using the Proposition 4.3 and the same technique in [Reitzner2011] (Lemma 4.6), we also obtain that if is a vector of U-statistics in the form such that are simple functions, then all kernels are also simple functions and

and

where stands for the set of partitions satisfying the conditions in Proposition 4.3 with and denotes the set of all partitions in of such that for any and any decomposition of into two disjoint sets there are and two variables which are in the same subset of .

By the formula of kernels in Proposition 4.1., we note that

This fact follows that

**Theorem 4.1.** *Assume that is a vector of U-statistics in the form (\ref{ustat}) such that are simple functions. Then
*

where and are defined in .

Now, we consider that a vector of -statistics in the form such that

Then, for each there exists a sequence such that and converges to -almost everywhere. Let give the vector of U-statistics , where

Hence,

Its follow that , converges to almost surely and all kernels in the Wiener-It\^o chaos expansion of are simple functions.

Note that

Moreover, the integrals

always exist for .

Therefore, by applying the Lebesgue dominated convergence theorem, we obtain that and for . Hence,

almost surely for .

Note that, the almost sure convergence implies the convergence in the probabilistic distance and , where is defined when we replace by in . Therefore, by using Theorem 4.1 and applying the triangular inequality, we conclude that

**Theorem 4.2.** *Assume that is a vector of U-statistics in the form such that
*

Then

where and are defined in .

**Corollary 4.3.** *Assume that is a sequence of vectors of U-statistics, which are defined as in Theorem 4.2, such that
*

for , then the law of converges to the multivariate Gaussian law .

**References**

[Bor96] Yu. V. Borovskikh, *U-statistics in Banach spaces*, VSP, Utrecht, 1996. MR1419498

[Hoe48] W. Hoeffding, *A class of statistics with asymptotically normal distribution*, Ann. Math. Statistics 19 (1948), 293–325. MR0026294

[HPA95] C. Houdr´e and V. P´erez-Abreu, *Covariance identities and inequalities for functionals on Wiener and Poisson spaces*, Ann. Probab. 23 (1995), no. 1, 400–419. MR1330776

[KB94] V. S. Koroljuk and Yu. V. Borovskich, *Theory of U-statistics*, Mathematics and its Applications, vol. 273, Kluwer Academic Publishers Group, Dordrecht, 1994. MR1472486

[Lee90] A. J. Lee, *U-statistics, theory and practice*, Statistics: Textbooks and Monographs, vol. 110, Marcel Dekker Inc., New York, 1990. MR1075417

[LP11] G. Last and M. Penrose, *Poisson process fock space representation, chaos expansion and covariance inequalities*, Probability Theory and Related Fields 150 (2011), 663–690.

[NV90] D. Nualart and J. Vives, *Anticipative calculus for the Poisson process based on the Fock space*, S´eminaire de Probabilit´es, XXIV, 1988/89, Lecture Notes in Math., vol. 1426, Springer, Berlin, 1990, pp. 154–165. MR1071538

[PSTU10] G. Peccati, J. L. Sol´e, M. S. Taqqu, and F. Utzet, *Stein’s method and normal approximation of Poisson functionals*, Ann. Probab. 38 (2010), no. 2, 443–478. MR2642882

[PT11] G. Peccati and M. S. Taqqu, *Wiener chaos: moments, cumulants and diagrams*, Bocconi & Springer Series, vol. 1, Springer, Milan, 2011. MR2791919

[PZ10] G. Peccati and C. Zheng, *Multi-dimensional Gaussian fluctuations on the Poisson space*, Electron. J. Probab. 15 (2010), no. 48, 1487–1527. MR2727319

[RS11] M. Reitzner and M. Schulte, *Central Limit Theorems for U-Statistics of Poisson Point Processes*, ArXiv e-prints (2011).

[Wu00] L. Wu, *A new modified logarithmic Sobolev inequality for Poisson point processes and several applications*, Probab. Theory Related Fields 118 (2000), no. 3, 427–438.

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