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## 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.

## Đạo hàm Malliavin

Xét không gian Hilbert khả tách $\mathcal H$ được trang bị tích vô hướng $\langle.,.\rangle$ và chuẩn $\|.\|$ tương ứng.  Thế thì tồn tại không gian xác suất $(\Omega, \mathcal{G},\mu)$ cùng với quá trình ngẫu nhiên $(W_h)_{h\in \mathcal H}$ tuyến tính theo quỹ đạo và ứng với mỗi $h$ cố định thì $W_h\in L^2(\Omega, \mathcal{G},\mu)$ là biến ngẫu nhiên Gauss, hơn nữa $\mathbf{E}(W_h)=0, \mathbf{cov}(W_{h_1}W_{h_2})=\langle h_1, h_2\rangle$. Xét $(e_1,e_2,...)$ là cơ sở trực chuẩn cố định của $\mathcal H$.

Thí dụ:

1. $\mathcal H=L^2([0,\infty)$ và tích phân Wiener $\displaystyle W_h=\int_0^{\infty}h(t)dW_t$ với $W_t$ là quá trình Wiener 1 chiều.

2. $\mathcal{H}=L^2(X, \mathcal{A}, m)$ là không gian độ đo $\sigma$-hữu hạn, và không tồn tại tập có đo dương không chứa thêm tập con nào nữa có độ đo dương bé hơn (tập hạt nhân). Ồn trắng $(W(A))_{A\in \mathcal{A}_f}$  ($\mathcal{A}_f$ là lớp tất cả các tập có độ đo hữu hạn, xem như là tập chỉ số) là một một quá trình Gauss  sao cho $W(A)\in L^2(\Omega, \mathcal G, \mu)$, $\mathbf{E}(W(A))=0$$\mathbf{cov}(W(A), W(B))=\mathcal{L}(A\cap B)$ ($\mathcal{L}$ – độ đo Lebesgue) và $W(A\cup B)=W(A)+W(B)$ nếu $A\cap B=\emptyset$. Với tập $A$ có độ đo hữu hạn, đặt $W(1_A)=W(A)$, từ đó mở rộng cho các hàm đơn giản trên $X$, và cuối cùng do tính trù, ta mật xây dựng được $W(h)$ với các hàm khả tích bậc hai $h\in L^2(X, \mathcal{A}, m)$.

Trở lại vấn đề của bài viết, kí hiệu $\mathcal P$ là lớp tất cả các biến ngẫu nhiên có dạng

$F=f(W(h_1),W(h_2),....,W(h_n)),\ h_1, h_2,...,h_n\in\mathcal{H}, \ n\ge 1$, với hàm trơn $f$ và các đạo hàm riêng của nó có độ tăng bậc đa thức. Dễ thấy $\mathcal P$ là tập con trù mật của $L^2(\Omega, \mathcal G, \mu)$.

Ta định nghĩa đạo hàm Malliavin của $F\in \mathcal P$ là biến ngẫu nhiên giá trị thuộc $\mathcal H$:

$\displaystyle DF=\sum_{k=1}^n\frac{\partial f}{\partial x_k}(W(h_1), W(h_2),...,W(h_n))h_k.$

Nó thỏa mãn quy tắc Leibnitz $\displaystyle D(FG)=FDG+GDF$.

Các tính chất đẹp:

1. $\displaystyle \mathbf{E}(\langle DF,h\rangle_{\mathcal{H}})=\mathbf{E}(FW(h))$

2. $\displaystyle \mathbf{E}(G\langle DF,h\rangle_{\mathcal{H}})=\mathbf{E}(-F\langle DG,h \rangle_{\mathcal{H}}+FGW(h))$

Vận dụng (2) bạn có thể chứng minh tính đóng được của toán tử $\displaystyle D$ từ $\displaystyle L^p(\Omega, \mathcal{G},\mu)$ vào $\displaystyle L^p(\Omega, \mathcal {H})$ với $\displaystyle p\ge 1$.

Với $\displaystyle p\ge 1$, kí hiệu $\displaystyle \mathbb{D}^{1,p}$ là bao đóng của $\mathcal P$ ứng với nửa chuẩn:

$\displaystyle \|F\|_{1,p}=\left(\mathbf{E}(|F|^p)+\mathbf{E}(\|DF\|_{\mathcal{H}}^p\right)^{1/p}$.

Đặt biệt với $\displaystyle p=2$, thì $\displaystyle \mathbb{D}^{1,2}$ xem như là không gian Hilbert với tích vô hướng:

$\displaystyle \langle F, G\rangle_{1,2}= \mathbf{E}(FG)+\mathbf{E}(\langle DF, DG\rangle_{\mathcal{H}})$.

Định nghĩa một cách đệ quy cho đạo hàm cấp cao $\displaystyle D^kF, F\in \mathcal{P}$ là vector ngẫu nhiên có giá trị thuộc không gian tích tensor $\displaystyle \mathcal{H}^{\otimes k}$. $D^k$ cũng là toán tử đóng được từ $L^p(\Omega, \mathcal{G}, \mu)$ vào $L^p(\Omega,\mathcal{H}^{\otimes k})$.

Với $\displaystyle k\in \mathbb{Z}_+, p\ge 1$, kí hiệu $\displaystyle \mathbb{D}^{k,p}$ là bao đóng của $\mathcal P$ ứng với nửa chuẩn:

$\displaystyle \|F\|_{k,p}=\left(\mathbf{E}(|X|^p)+\sum_{l=1}^k\mathbf{E}(\|D^lF\|^p_{\mathcal{H}^{\otimes k}}\|)\right)^{1/p}$.

Kí hiệu

$\displaystyle \mathbb{D}^{\infty}=\bigcap_{k,p} \mathbb{D}^{k,p}$.

Chú ý là với $\displaystyle k \ge 1, p > q$ ta có quan hệ lồng nhau: $\displaystyle \mathbb{D}^{k,p}\subset \mathbb{D}^{k-1,q}.$

Bằng cách lấy giới hạn, ta có thể xác định đạo hàm Malliavin $D^kF$ với $\displaystyle F\in\mathbb{D}^{k,p}\subset L^p(\Omega, \mathcal{G},\mu)$ tương ứng.

Đạo hàm Malliavin thỏa mãn luật xích theo nghĩa: Cho $\displaystyle \phi:\mathbb{R}^n\to \mathbb{R}$ khả vi liên tục với các đạo hàm riêng bị chặn, các biến ngẫu nhiên $\displaystyle F_1,F_2,...,F_n\in \mathbb{D}^{1,p}$ thế thì $\displaystyle \phi(F_1,F_2,...,F_n)\in \mathbb{D}^{1,p}$

$\displaystyle D(\phi(F_1,F_2,...,F_n))=\sum_{k=1}^n \frac{\partial \phi}{\partial x_i}(F_1,F_2,...,F_n)DF_i$.

Với $\displaystyle F\in \mathbb{D}^{k,p}, G\in \mathbb{D}^{k,q}, k\in \mathbb{Z}_+, 1$\displaystyle \frac{1}{r}=\frac{1}{p}+\frac{1}{q}$ thế thì $\displaystyle FG\in \mathbb{D}^{k,r}$ và bất đẳng thức loại Holder sau được thỏa mãn

$\displaystyle\|FG\|_{k,r}\le C(p,q,k)\|F\|_{k,p}\|G\|_{k,q},$

ở đây $\displaystyle C(p,q,k)$ là hằng số nào đó chỉ phụ thuộc $\displaystyle p,q,k$.

## Toán tử Ornstein-Uhlenbeck

Trường hợp hữu hạn chiều

Cho không gian xác suất $(\mathbb{R}^m, \mathfrak{B}(\mathbb{R}^m), \mu )$ với $\mathfrak{B}(\mathbb{R}^m)$$\sigma$-đại số Borel trên $\mathbb{R}^m$$\mu$ là độ đo Gauss:

$\displaystyle\mu(dx)=\frac{1}{(2\pi)^{m/2}} e^{-|x|^2/2}dx.$

Xét phương trình vi phân ngẫu nhiên

$\displaystyle dX_t=\sqrt{2}dW_t-X_tdt$, với $W_t$ là quá trình Wiener trong $\mathbb{R}^m.$

Áp dụng công thức Ito thế thì

$\displaystyle X_t(x)=e^{-t}x+\sqrt{2}\int_0^t e^{-(t-s)}dW_s.$

Ta định nghĩa toán tử $P_t$ xác định trên $L^p(\mathbb{R}^m, \mu), p\ge 1$

$\displaystyle P_t f(x)=\mathbf{E}(f(X_t(x))=\int_{\mathbb{R}^m} f(e^{-t}x+\sqrt{1-e^{-2t}}y)\mu(dy), \ t\ge 0$.

Các tính chất đẹp:

1. $P_t$ là toán tử nửa nhóm trên $L^p(\mathbb{R}^m, \mu)$

2. $\displaystyle \| P_tf(x)\|_{L^p(\mathbb{R}^m, \mu)} \le \| f\|_{L^p(\mathbb{R}^m, \mu)}, p\ge 1$

3. $P_t$ là toán tử đối xứng trên $L^2(\mathbb{R}^m, \mu)$

4.  $P_t$ thu hẹp trên $C_b^2(\mathbb{R}^m)$ có  infinitesimal generator là $L_m=\Delta-x.\nabla$

Mở rộng trên không gian Hilbert khả tách

Giả sử không gian Hilbert khả tách $\mathcal H$ ứng với tích vô hướng $\langle.,.\rangle$ và chuẩn $\|.\|$ tương ứng.  Thế thì tồn tại không gian xác suất $(\Omega, \mathcal{G},\mu)$ cùng với quá trình ngẫu nhiên $(W_h)_{h\in \mathcal H}$ tuyến tính theo quỹ đạo và ứng với mỗi $h$ cố định thì $W_h$ là biến ngẫu nhiên Gauss hơn nữa $\mathbf{E}(W_h)=0, \mathbf{cov}(W_{h_1}W_{h_2})=\langle h_1, h_2\rangle$. Xét $(e_1,e_2,...)$ là cơ sở trực chuẩn của $\mathcal H$.

Trên không gian $L^p(\Omega, \mu), p\ge 1$ các biến ngẫu nhiên khả tích bậc $p$, xác định toán tử

$\displaystyle P_t F=\int_{\Omega}F(e^{-t}\omega +\sqrt{1-e^{-2t}}\chi)\mu(d\chi), \ t\ge 0$.

Các tính chất (1-2-3) trong trường hợp hữu hạn chiều $P_t$ vẫn đúng trên $(\Omega, \mathcal{G},\mu)$.

Với bộ chỉ số $a=(a_1,a_2,...),\ a_i\in \mathbb{Z_+}$, đặt

trong đó sử dụng kí hiệu đa thức Hermite
$\displaystyle H_n(x)=\frac{1}{n!}\frac{d^n}{dt^n}\left. e^{-t^2/2+tx}\right|_{t=0}$.

Không khó khăn để kiểm tra $(H_a)$ lập thành cơ sở trực chuẩn của $L^2(\Omega,\mathcal G, \mu)$.

Kí hiệu $\mathcal{W}_n$ là không gian con đóng của không gian Hilbert $L^2(\Omega,\mathcal G, \mu)$ sinh bởi hệ trực chuẩn $(H_a, \sum_{k=1}^{\infty}{|a_k|}=n)$. Khi đó ta có biểu diễn hỗn độn Wiener
$L^2(\Omega,\mathcal G, \mu)=\bigoplus_{n=0}^{\infty} \mathcal{W}_n$
và không gian $\mathcal{W}_n$ gọi là hỗn độn Wiener thứ $\displaystyle n$.

Toán tử $P_t$ được phân tích theo các toán tử chiếu trực giao $\displaystyle J_n$ từ $\displaystyle L^2(\Omega,\mathcal G, \mu)$ xuống  $\displaystyle \mathcal{W}_n$ như sau


với $F\in L^2(\Omega,\mathcal G, \mu)$

Ta xác định được



là infinitesimal generator của toán tử nửa nhóm $P_t$ thu hẹp trên miền


$L$ được gọi là toán tử Ornstein-Uhlenbeck, nó cùng với đạo hàm Malliavin và tích phân Skorohod là 3 toán tử nền tảng nhất của ngành Biến phân ngẫu nhiên.