Archive for the ‘Stochastic Calculus of Variations’ Category

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

November 17, 2011 Leave a comment

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




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 (E, \mathcal A, \mu) be some measure space with \sigma-finite measure \mu. The Poisson point process or Poisson measure with intensity measure \mu is a family of random variables \{{N}(A)\}_{A\in\mathcal{A}} defined on some probability space (\Omega, \mathcal F, \mathrm{P}) such that
1.  \forall A\in\mathcal{A}, N(A) is a Poisson random variable with rate \mu(A).
2. If sets A_1,A_2,\ldots,A_n\in\mathcal{A} don’t intersect then the corresponding random variables {N}(.) from i) are mutually independent.
3. \forall\omega\in\Omega, {N}(.,\omega) is a measure on (E, \mathcal A).

Note that, for some \sigma-finite measure space (E, \mathcal A, \mu), we can always set

\Omega=\{\omega =\sum_{j=1}^n\delta_{z_j}, \ n\in\mathbb{N}\cup\{\infty\}, z_j\in E  \},

where \delta_z denotes the Dirac mass at z, and for each A\in \mathcal{A}, we give the mapping {N} such that

\omega\mapsto {N}(A,\omega)=\omega(A).

Moreover, the \sigma-field \mathcal{F} is supposed to be the \text{P}-completion of the \sigma-field generated by {N}.

Let \mathcal{M}(E) denote the space of all integer-valued \sigma-finite measures on E, which can be equipped with the smallest \sigma-algebra \Sigma such that for each A\in\mathcal{A} then the mapping \eta\in\mathcal{M}(E)\mapsto \eta(A) is measurable. We can give on (\mathcal{M}(E),\Sigma) the probability measure P_{N} induced by the Poisson measure N and denote L^p(P_{N}) as the set of all measurable functions F:\mathcal{M}(E)\to \overline{\mathbb R} such that \mathbf{E}[|F|^p]<\infty, where the expectation takes w.r.t. the probability measure P_{N}.

Let L^p(\mu^n) be the space of all measurable functions f: E^n \to\overline{\mathbb{R}} such that

\|f\|=\int\limits_{E^k}|f(z_1,\hdots,z_n)|^p \, \mu^n(dz_1, \dots ,dz_n)<\infty.

Note that L^2(\mu^n) becomes a Hilbert space when we define on it the scalar product

 \langle f, g\rangle =\int\limits_{E^k}f(z_1,\hdots,z_n) g(z_1,\hdots,z_n)\, \mu^n(dz_1, \dots ,dz_n).

We denote L^p_{\text{sym}}(\mu^n) as the subset of symmetric functions in L^p(\mu^n), in the sense that the functions are invariant under all permutations of the arguments. Now, for each A\in\mathcal{A}, we define the random variable \widehat{N}(A)=N(A)-\mu(A), which is also known as a compensated Poisson measure.
For each symmetric function f\in L^2_{\text{sym}}(\mu^n), one can define the multiple Wiener-It\^o integral I_n(f) w.r.t the compensated Poisson measure \widehat{N} denoted by

I_n(f)=\int_{E^n}f(z_1,\hdots, z_n) \widehat{N}^n(dz_1, \dots ,dz_n). \ \ \ (1)

At first, let \mathcal{S}_n be the class of simple functions f, which takes the form

f(z_1,z_2,....,z_n)=\sum_{k=1}^m\lambda_k \mathbf{1}_{A_{1}^{(k)}\times\ldots \times A_{n}^{(k)}}(z_1,z_2,....,z_n), \ \ (2)

where A_{i}^{(k)}\in\mathcal{A}, \lambda_k\in \mathbb{R} and the sets A_{1}^{(k)}\times\ldots \times A_{n}^{(k)} are pairwise disjoint such that f is symmetric and vanishes on diagonals, that means f(z_1, \ldots , z_n) = 0 if z_i = z_j for some i\neq j. The multiple Wiener-It\^o integral for a simple function f in the form (2) with respect to the compensated Poisson measure \widehat{N} is defined by


Since the class \mathcal{S}_n is dense in L^2_{\rm sym}(\mu^n) then for every f\in L^2_{\rm sym}(\mu^n), there exists a sequence \{f_l\}_{l\ge0}\subset\mathcal{S}_n such that f_l\to f in L^2_{\rm sym}(\mu^n). Moreover, one can show that \mathbf{E}[I_n(f_l)^2] = k!\|f_l\|^2. Hence, the the multiple Wiener-It\^o integral I_n(f) for a symmetric function f\in L^2_{\rm sym}(\mu^n) can be defined as the limit of the sequence \{I_n(f_l)\}_{l\ge 0} in L^2(P_N) and we denote it as (1).

Proposition 2.1. For n,m\geq 1 and f\in L_{\rm sym}^2(\mu^n), g\in L_{\rm sym}^2(\mu^m), then
1. \mathbf{E}[I_n(f)]=0,
2. \mathbf{E}[I_n(f) I_m(g)]=  \delta_{n,m} n!\langle f,g  \rangle_{L^2(\mu^n)}
where \delta_{n,m} is the Kronecker delta

For a measurable function F:\mathcal{M}(E)\to\overline{\mathbb R} and z\in E we define the difference operator as


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


We define the kernels of F as functions f_n: E^n\to \overline{\mathbb R} given by

f_n(z_1,\hdots,z_n)=\frac{1}{n!}\mathbf{E}[D_{z_1,\hdots,z_n}F], n \geq 1,

Note that f_n is a symmetric function.

We define the Ornstein-Uhlenbeck generator as

LF(\eta) = \int\limits_E (F( \eta - \delta_z) - F(\eta)) \eta(dz) - \int\limits_E (F(\eta) - F(\eta + \delta _z))\,  \mu(dz).

Proposition 2.2. For each F\in L^2(P_N), then the kernels f_n are elements of L^2(\mu^n), n\ge 1 and uniquely admit the Wiener-It\^o chaos expansion in the form


where the sum converges in L^2(P_N). Furthermore, for F,G\in L^2(P_N)

{\rm Cov}(F,G)=\mathbf{E}[FG]- \mathbf{E}[F]\mathbf{E}[G]=\sum_{n=1}^\infty n!\langle f_n, g_n \rangle_{L^2(\mu^n)}.

Proposition 2.3. Let F\in L^2(P_N), and assume that

\sum_{n=1}^\infty n \, n!\|f_n\|^2<\infty.

Then the difference of F at z\in E is given by

D_zF=\sum_{n=1}^\infty n I_{n-1}(f_n(y,\cdot)).

Proposition 2.4. For each random variable F\in L^2(P_N) such that

\sum_{n=1}^\infty i^2i!\|f_n\|^2<\infty,

then the Ornstein-Uhlenbeck generator L is calculated as

LF=-\sum_{n=1}^{\infty}n I_n(f_n).

Moreover, its inverse operator is calculated as


for each F\in L^2(P_N) such that \mathbf{E}[F]=0.

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 d-dimensional random vectors X,Y such that \mathbf{E}(\|X\|_{\mathbb{R}^d}), \mathbf{E}(\|Y\|_{\mathbb{R}^d})<\infty, which is defined by

\Delta(X,Y)=\sup_{g\in \mathcal H}|\mathbf{E}(g(X))-\mathbf{E}(g(Y))|,

where  \mathcal{H} is the family of all real-valued functions g\in C^2(\mathbb{R}^d) such that

\|g\|_{{\rm Lip}}=\sup_{x\neq y}\frac{|g(x)-g(y)|}{\|x-y\|_{\mathbb{R}^d}}\le 1, \sup_{x\in\mathbb{R}^d}\| {\rm Hess}( g(x))\|\le 1.

In the above inequality,

{\rm Hess}(g(z))=\left.\left(\frac{\partial^2 g}{\partial x_i\partial x_j}(z)\right)_{i,j=1}^d\right.

stands for the Hessian matrix of g evaluated at a point z and we use the notation of operator norm for a d\times d real matrix A given by

\|A \| = \sup_{\|x\|_{\mathbb{R}^d}=1} \|Ax\|_{\mathbb{R}^d}.

Theorem 3.1.
[Multi-dimensional Malliavin-Stein inequality] Consider a random vector F=(F_1,\ldots,F_d)\subset L^2(P_N), d\ge2 such that for 1\leq i\leq d, F_i\in{\rm dom}(D), and \mathbf{E}(F_i)=0. Suppose that  X\sim\mathcal{N}_d(0,C) , where C=\{C(i,j): i,j= 1,\ldots,d  \} is a d\times d positive definite symmetric matrix. Then,

\begin{array}{ll}\displaystyle\Delta(F,X) \leq \|C^{-1}\| \|C\|^{1/2} \sqrt{\sum_{i,j=1}^{d} \mathbf{E}[(C(i,j) - \langle  DF_i,-DL^{-1}F_j \rangle_{L^2(\mu)} )^2 ] } \\\displaystyle+ \cfrac{\sqrt{2\pi}}{8} \|C^{-1}\|^{3/2} \|C\| \int_E \mu(dz)\mathbf{E}\left[\left(\sum_{i=1}^d|D_z F_i | \right)^2 \left(\sum_{i=1}^d|D_z L^{-1} F_i |  \right) \right].\end{array}

For the proof, we refer to [Peccati2010b].

Now we consider a d-dimensional random vector F=(F_1,\ldots,F_d) \subset L^2(P_N) with the covariance matrix \Sigma=\{\Sigma(i,j): i,j= 1,\ldots,d  \} such that each component F_i has finite Wiener-It\^o chaos expansions with kernels f_{i}^{(n)}, which vanishes if n>k. Let give the centered random vector


where \sqrt{A} stands for the square root of a positive definite matrix A, i.e if A has the eigenvalues decomposition A= P^{-1}{\rm diag}({\lambda_1},\ldots,{\lambda_d})P

\sqrt{A}=P^{-1}{\rm diag}(\sqrt{\lambda_1},\ldots,\sqrt{\lambda_d})P.

Let us use vector notations

\nabla_zF=(D_zF_1,\ldots,D_zF_d), \ \nabla_z L F=(D_z LF_1,\ldots ,D_z LF_d)

and note that the inequality

{\rm trace}(AB)\le {\rm trace}(A) {\rm trace}(B)

holds for all positive definite matrices A,B.
Therefore, by the properties matrix trace {\rm trace}(AB)={\rm trace}(BA) and using the inequality (3), we have

\sum_{i,j=1}^{d}(C(i,j) - \langle  DG_i,-DL^{-1}G_j \rangle_{L^2(\mu)} )^2 ={\rm trace}\left[\left(C- \int_E\mu(dz) \nabla_z G(-\nabla_z L G)^T\right)^2\right]
={\rm trace}\left[\left(\sqrt{C\Sigma^{-1}}\left( \Sigma - \int_E\mu(dz) \nabla_z [F-\mathbf{E}(F)](-\nabla_z L [F-\mathbf{E}(F)])^T\right)\sqrt{C\Sigma^{-1}}^T\right)^2\right]
={\rm trace}\left[ (C\Sigma^{-1})^2 \left( \Sigma - \int_E\mu(dz) \nabla_z [F-\mathbf{E}(F)](-\nabla_z L [F-\mathbf{E}(F)])^T\right)^2\right]
\le {\rm trace} [(C\Sigma^{-1})^2]{\rm trace} \left[\left( \Sigma - \int_E\mu(dz) \nabla_z [F-\mathbf{E}(F)](-\nabla_z L [F-\mathbf{E}(F)])^T\right)^2\right]
=\|C\Sigma^{-1}\|_{F}^2 \sum_{i,j=1}^{d}\left(\Sigma(i,j) - \langle  D[F_i-\mathbf{E}(F)],-DL^{-1}[F_j-\mathbf{E}(F)] \rangle_{L^2(\mu)} \right)^2,


\|A\|_F=\sqrt{{\rm trace}(A^TA)}

denotes the Frobenius norm of matrix A.
Note that

\Sigma(i,j)={\rm Cov}(F_i,F_j)=\sum_{n=1}^{k}n!\left\langle f_i^{(n)}, f_j^{(n)} \right\rangle_{L^2(\mu^n)},


 \langle  D[F_i-\mathbf{E}(F_i)],-DL^{-1}[F_j-\mathbf{E}(F_j)] \rangle_{L^2(\mu)}=\left \langle \sum_{n=1}^knI_{n-1}(f_i^{(n)}(z,\cdot)),\sum_{n=1}^k I_{n-1}(f_j^{(n)}(z,\cdot))\right\rangle_{L^2(\mu)}.


\begin{array}{l} \displaystyle\mathbf{E}[(\Sigma(i,j) - \langle  D[F_i-\mathbf{E}(F_i)],-DL^{-1}[F_j-\mathbf{E}(F_j)] \rangle_{L^2(\mu)} )^2]\\ \displaystyle\le k^2\left(  \sum_{n=1}^{k}\mathbf{E}\left[\left(n!\left\langle f_i^{(n)}, f_j^{(n)} \right\rangle_{L^2(\mu^n)}-n\left \langle I_{n-1}(f_i^{(n)}(z,\cdot)),I_{n-1}(f_j^{(n)}(z,\cdot)) \right\rangle_{L^2(\mu)}\right)^2 \right]\right.\\  \displaystyle +\left. \sum_{n,m=1,\, n \neq m}^k  \mathbf{E} \left[n^2 \left\langle I_{n-1}(f_i^{(n)}(z,\cdot)),I_{m-1}(f_j^{(m)}(z,\cdot))\right\rangle_{L^2(\mu)}^2\right]\right)\\  \displaystyle =\ k^2 \sum_{1\le n,m\le k}n^2{\rm Var}\left( \left \langle I_{n-1}(f_i^{(n)}(z,\cdot)),I_{m-1}(f_j^{(m)}(z,\cdot)) \right\rangle_{L^2(\mu)} \right).\end{array}

It follows that

\sqrt{\sum_{i,j=1}^{d} \mathbf{E}[(C(i,j) - \langle  DG_i,-DL^{-1}G_j \rangle_{L^2(\mu)} )^2 ] }
\le k^2 \|C\Sigma^{-1}\|_{F} \sqrt{\sum_{i,j=1}^{d}\sum_{n,m=1}^{k} {\rm Var}\left( \left \langle I_{n-1}(f_i^{(n)}(z,\cdot)),I_{m-1}(f_j^{(m)}(z,\cdot)) \right \rangle_{L^2(\mu)} \right) }. \ \ (4)

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

 \int_E \mu(dz)\mathbf{E}\left[\left(\sum_{i=1}^d|D_z G_i | \right)^2 \left(\sum_{i=1}^d|D_z L^{-1} G_i |  \right) \right]\le d^{3/2}\int_E \mu(dz)\mathbf{E}\left[\|\nabla_z G \|_{\mathbb{R}^d}^2\|\nabla_zL^{-1} G \|_{\mathbb{R}^d} \right]
=d^{3/2}\int_E \mu(dz)\mathbf{E}\left[\| \sqrt{C\Sigma^{-1}} \nabla_z [F-\mathbf{E}(F)]\|_{\mathbb{R}^d}^2\|\sqrt{C\Sigma^{-1}} \nabla_zL^{-1}[F-\mathbf{E}(F)]  \|_{\mathbb{R}^d} \right]
\le d^{3/2}\|\sqrt{C\Sigma^{-1}}\|^3\int_E \mu(dz)\mathbf{E}\left[\|  \nabla_z [F-\mathbf{E}(F)]\|_{\mathbb{R}^d}^2\| \nabla_zL^{-1} [F-\mathbf{E}(F)] \|_{\mathbb{R}^d} \right]
\le d^{3/2}\|\sqrt{C\Sigma^{-1}}\|^3 \left( \int_E \mu(dz) \mathbf{E}\left[\left(\sum_{i=1}^d|D_z [F_i-\mathbf{E}(F_i)] |^2\right)^2\right]  \right)^{1/2}
\times \left( \int_E \mu(dz) \mathbf{E}\left[ \sum_{i=1}^d|D_z L^{-1} [F_i-\mathbf{E}(F_i)] |^2   \right]  \right)^{1/2}

\le d^2 \|\sqrt{C\Sigma^{-1}}\|^3  \left( \sum_{i=1}^d \int_E \mu(dz) E\left[|D_z [F_i-\mathbf{E}(F_i)] |^4\right]  \right)^{1/2}

\times \left( \sum_{i=1}^d \int_E \mu(dz) \mathbf{E}\left[|D_z L^{-1} [F_i-\mathbf{E}(F_i)] |^2\right]  \right)^{1/2}

\le  d^2 \|\sqrt{C\Sigma^{-1}}\|^3  \left( \sum_{i=1}^d   \int\limits_E\mu(dz) k^3 \sum_{n=1}^k  n^4 \mathbf{E}[I_{n-1}(f_i^{(n)}(z,\cdot))^4] \right)^{1/2}

\times \left( \sum_{i=1}^d                 \int\limits_E\mu(dz)\sum_{n=1}^k \mathbf{E}[I_{n-1}(f_i^{(n)}(z,\cdot))^2]\right)^{1/2}

=  d^2  \|\sqrt{C\Sigma^{-1}}\|^3 \left( \sum_{i=1}^d  k^3 \sum_{n=1}^k n^4 \mathbf{E}[\|I_{n-1}(f_i^{(n)}(z,\cdot))^2 \|^2] \right)^{1/2}  \left( \sum_{i=1}^d \sum_{n=1}^k  (n-1)!\|f_i^{(n)}\|^2  \right)^{1/2}

\le d^{2} k^{7/2} \|\sqrt{C\Sigma^{-1}}\|^3({\rm trace}(\Sigma))^{1/2} \sqrt{\sum_{i=1}^d\sum_{n=1}^k \mathbf{E}\left[\|I_{n-1}(f_i^{(n)}(z,\cdot))^2 \|^2\right]}. \ \ (5)
Substituting (4) and (5) to the inequality in Theorem 3.1 for G, we obtain that

Theorem 3.2. Let give a d-dimensional Gaussian random variable  X\sim \mathcal{N}_d(0,C). Assume that F=(F_1,\ldots,F_d) \subset L^2(P_N) such that {\rm Cov}(F_i,F_j)=\Sigma(i,j), \ i,j=1,d and F_i has finite Wiener-It\^o chaos expansions with kernels f_{i}^{(n)}, which vanishes if n>k. Then

\begin{array}{lll}\displaystyle\Delta\left(\sqrt{C\Sigma^{-1}}\left(F-\mathbf{E}(F)\right),X\right) \leq \\ \displaystyle  \cfrac{ \sqrt{2\pi}}{8}d^{2} k^{7/2} \|\sqrt{C\Sigma^{-1}}\|^3 \|C^{-1}\|^{3/2} \|C\| ({\rm trace}(\Sigma))^{1/2} \sqrt{\sum_{i=1}^d\sum_{n=1}^k \mathbf{E}\left[\|I_{n-1}(f_i^{(n)}(z,\cdot))^2 \|^2\right]}\\ \displaystyle +  k^2\|C\Sigma^{-1}\|_{F} \|C^{-1}\| \|C\|^{1/2}\sqrt{\sum_{i,j=1}^{d}\sum_{n,m=1}^{k} {\rm Var}\left( \left \langle I_{n-1}(f_i^{(n)}(z,\cdot)),I_{n-1}(f_j^{(m)}(z,\cdot)) \right \rangle_{L^2(\mu)} \right) }.\end{array}
IV – Application for multi-dimensional U-statistics

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

 F = \left(\sum_{({z}_1,\ldots,{z}_{k_1}) \in S_{k_1}(N)} \phi_1({z}_1,\dots,{z}_{k_1}),\ldots,\sum_{({z}_1,\dots,{z}_{k_d}) \in S_{k_d}(N)} \phi_d({z}_1,\dots,{z}_{k_d}) \right), \ \ (6)

where \phi_i\in L^1_{\rm sym}(\mu^{k_i}), and S_{k_i}(N) denotes the set of all k_i-tuples of distinct points of N. This means that each component

F_i=\sum_{({z}_1,\dots,{z}_{k_i})\in S_{k_i}(N)} \phi_i({z}_1,\dots,{z}_{k_i})

is an U-statistic of order k_i with respect to the Poisson point process N, i=1,d.

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

Proposition 4.1. Let F\in L^2(P_N) be a U-statistic of order k in the form

F=\sum_{({z}_1,\dots,{z}_{k})\in S_k(N)} \phi({z}_1,\dots,{z}_k)

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

 f_n(z_1,\hdots,z_n)= \begin{cases}\displaystyle \binom{k}{n}\int\limits_{E^{k-n}}\phi(z_1,\hdots,z_n,x_1,\hdots,x_{k-n})\, \mu^{k-n}(dx_1,\dots,dx_{k-n}), &n\leq k\\ 0, & n>k. \end{cases}

Proposition 4.2. Assume F\in L^2(P_N), then
1. If F is a U-statistic, then F has a finite Wiener-It\^o chaos expansion with kernels f_n\in L^1(\mu^n)\cap L^2(\mu^n) , n=1,\hdots,k.
2. If F has a finite Wiener-Itô chaos expansion with kernels f_n\in L^1(\mu^n) \cap L^ 2 (\mu^n), n=1,\hdots,k, then F is a finite sum of U-statistics and a constant.

Proposition 4.3. Let f_i\in \mathcal{S}_{k_i}, \ i=1,\hdots,m and  \Pi be the set of all partitions of Z=\{z_1^{(1)},\dots,z_{n_1}^{(1)},\dots,z_{1}^{(m)} ,\dots, z_{n_m}^{(m)}\}, n_i\le k_i such that for each \pi \in \Pi,
1. z^{(i)}_{l}, z^{(i)}_{h}\in Z, l\neq h are always in different subsets of \pi, and such that
2. every subset of \pi has at least two elements.
For every partition \pi\in\Pi we define an operator R^{\pi} that replaces all elements of Z in \prod_{i=1}^m f_i(z_1^{(i)},\hdots,z_{n_i}^{(i)}) that belong to the same subset of \pi by a new variable x_j, j =1, \hdots,{|\pi|}, where |\pi| denotes the number of subsets of the partition \pi. Then

\mathbf{E} \left[\prod_{i=1}^m I_{n_i}(f_i)\right]=\sum_{\pi\in\Pi}\int\limits_{E^{|\pi|}}R^{\pi}(\prod_{i=1}^m f_i(\cdot))(x_1,\hdots,x_{|\pi|})\, \mu^{|\pi|}(dx_1,\dots,dx_{|\pi|}).

Using the Proposition 4.3 and the same technique in [Reitzner2011] (Lemma 4.6), we also obtain that if F=(F_1,F_2,...,F_d)\subset L^2(P_N) is a vector of U-statistics in the form (6) such that \phi_i, i=1,d are simple functions, then all kernels f_i^{(n)} are also simple functions and

{\rm Var}\left( \left \langle I_{n-1}(f_i^{(n)}(z,\cdot)),I_{m-1}(f_j^{(m)}(z,\cdot)) \right \rangle_{L^2(\mu)} \right)
= \mathbf{E}\left[ \left \langle I_{n-1}(f_i^{(n)}(z,\cdot)),I_{m-1}(f_j^{(m)}(z,\cdot)) \right \rangle_{L^2(\mu)}^2\right]- \delta_{n,m}\left((n-1)!\left\langle f_i^{(n)}, f_j^{(m)} \right\rangle_{L^2(\mu^n)}\right)^2
= \int_{E^2}\mathbf{E}\left[ I_{n-1}(f_i^{(n)}(z,\cdot))^2 I_{m-1}(f_j^{(m)}(y,\cdot))^2 \right] \mu^2(dy,dz)
-\delta_{n,m}\left((n-1)!\left\langle f_i^{(n)}, f_j^{(m)} \right\rangle_{L^2(\mu^n)}\right)^2
\le \sum_{\pi\in \overline{\Pi}_{n,m}}\int_{E^{|\pi|}}R^{\pi}\left( \left|f^{(n)}_i(.)f^{(n)}_i(.)f^{(m)}_j(.)f^{(m)}_j(.)\right|\right)(x_1,\ldots, x_{|\pi|})\mu^{|\pi|}(dx_1,\ldots, dx_{|\pi|}),


\mathbf{E}\left[\|I_{n-1}(f_i^{(n)}(z,\cdot))^2 \|^2\right]=\int_{E}\mathbf{E}\left[I_{n-1}(f_i^{(n)}(z,\cdot))^4\right]\mu(dz)

\le \sum_{\pi\in \overline{\Pi}_{n,m}}\int_{E^{|\pi|}}R^{\pi}\left( \left|f^{(n)}_i(.)f^{(n)}_i(.)f^{(n)}_i(.)f^{(n)}_i(.)\right|\right)(x_1,\ldots, x_{|\pi|})\mu^{|\pi|}(dx_1,\ldots, dx_{|\pi|}),
where  \Pi_{n,n} stands for the set of partitions satisfying the conditions in Proposition 4.3 with Z_{n,m}=\{z_{1}^{(1)}\ldots z_{n-1}^{(1)},z_{1}^{(2)}\ldots z_{n-1}^{(2)},z_{1}^{(3)}\ldots z_{m-1}^{(3)},z_{1}^{(4)}\ldots z_{m-1}^{(4)}\} and \overline{\Pi}_{n,m} \subset \Pi_{n,m} denotes the set of all partitions in \Pi_{n,m} of such that for any \pi \in \overline{\Pi}_{n,m} and any decomposition of \{ 1, 2,3,4\} into two disjoint sets M_1, M_2 there are i \in M_1, j \in M_2 and two variables z_l^{(i)}, z_h^{(j)} which are in the same subset of \pi.
By the formula of kernels in Proposition 4.1., we note that

\sum_{\pi\in \overline{\Pi}}\int_{E^{|\pi|}}R^{\pi}\left( \left|f^{(n)}_i(.)f^{(n)}_i(.)f^{(m)}_j(.)f^{(m)}_j(.)\right|\right)(x_1,\ldots, x_{|\pi|})\mu^{|\pi|}(dx_1,\ldots, dx_{|\pi|})
\le M_{n,m}(i,j) = \mathbf{1}_{n\le k_i, m\le k_j} \binom{k_i}{n}^2\binom{k_j}{m}^2  \sum_{\pi \in \overline{\Pi}_{n,m}}\int\limits_{E^{|\pi|}} \int\limits_{E^{2(k_i-m)}}   \int\limits_{E^{2(k_j-n)}}
R^{\pi} \left(  |\prod_{l=1}^2 \phi_i(\cdot , x^{(l)}_1,\hdots,x^{(l)}_{k_i-n})\prod_{l=3}^4 \phi_j(\cdot , x^{(l)}_1,\hdots,x^{(l)}_{k_j-m})| \right)(y_1,\hdots,y_{|\pi|}),
\mu^{|\pi|+2(k_i+k_j-j-i)}(dx_1^{(1)},\hdots,dx_{k_i-n}^{(2)},dx_{1}^{(3)} \ldots dx_{k_j-m}^{(4)}, dy_1, \dots dy_{|\pi|}). \ \ (7)

This fact follows that

Theorem 4.1. Assume that F=(F_1,F_2,...,F_d)\subset L^2(P_N) is a vector of U-statistics in the form (\ref{ustat}) such that \phi_i, i=1,d are simple functions. Then

\begin{array}{lll}\displaystyle\Delta\left(\sqrt{C\Sigma^{-1}}\left(F-\mathbf{E}(F)\right),X\right) \leq \\ \displaystyle  \cfrac{ \sqrt{2\pi}}{8}d^{2} k^{7/2} \|\sqrt{C\Sigma^{-1}}\|^3 \|C^{-1}\|^{3/2} \|C\| ({\rm trace}(\Sigma))^{1/2} \sqrt{\sum_{i=1}^d\sum_{n=1}^k M_{n,n}(i,i)}\\ \displaystyle +  k^2\|C\Sigma^{-1}\|_{F} \|C^{-1}\| \|C\|^{1/2}\sqrt{\sum_{i,j=1}^{d}\sum_{n,m=1}^{k} M_{n,m}(i,j) },\end{array}

where k=\max\{k_i, 1\le i\le,d\} and M_{n,m}(i,j), 1\le i,j\le d, 1\le n,m\le k are defined in (7).

Now, we consider that F=(F_1,F_2,....,F_d)\subset L^2(P_N) a vector of U-statistics in the form (6) such that

\sum_{({z}_1,\dots,{z}_{k_i})\in S_{k_i}(N)} |\phi_i({z}_1,\dots,{z}_{k_i})|\in L^2(P_N).

Then, for each i=1,2,\ldots,d there exists a sequence \{\phi_{i,l}\}_{l\ge 0}\subset \mathcal{S}_{k_i} such that |\phi_{i,l}|\le |\phi_i| and \phi_{i,l} converges to \phi_i \mu^{k_i}-almost everywhere. Let give the vector of U-statistics F^{(l)}=(F_{1,l},\ldots,F_{d,l}), where

F_{i,l}=\sum_{({z}_1,\dots,{z}_{k_i})\in S_{k_i}(N)} \phi_{i,l}({z}_1,\dots,{z}_{k_i}).


|F_{i,l}|\le \sum_{({z}_1,\dots,{z}_{k_i})\in S_{k_i}(N)} |\phi_{i,l}({z}_1,\dots,{z}_{k_i})|\le \sum_{({z}_1,\dots,{z}_{k_i})\in S_{k_i}(N)} |\phi_i({z}_1,\dots,{z}_{k_i})|\in L^2(P_N).

Its follow that F_{i,l}\in L^2(P_N), F_{i,l} converges to F_i almost surely and all kernels f_{i,l}^{(n)} in the Wiener-It\^o chaos expansion of F_{i,l} are simple functions.
Note that

\Sigma(i,j)={\rm Cov}(F_i,F_j)= =\sum_{n=1}^{\infty}n!\binom{k_i}{n}\binom{k_j}{n}\int\limits_{E^n}\ \int\limits_{E^{k_i-n}} \phi_i(z_1,\hdots,z_n,x_1,\hdots,x_{k_i-n})\mu^{k_i-n}(dx_1, \dots dx_{k_i-n}) \times \int\limits_{E^{k_j-n}}\phi_j(z_1,\hdots,z_n,x_1,\hdots,x_{k_j-n})\, \mu^{k_j-n}(dx_1, \dots dx_{k_j-n}) \mu^n(dz_1, \dots, dz_n).

Moreover, the integrals

\int\limits_{E^n}\ \int\limits_{E^{k_i-n}} |\phi_i(z_1,\hdots,z_n,x_1,\hdots,x_{k_i-n})|\mu^{k_i-n}(dx_1, \dots dx_{k_i-n}) \times \int\limits_{E^{k_j-n}}|\phi_j(z_1,\hdots,z_n,x_1,\hdots,x_{k_j-n})|\, \mu^{k_j-n}(dx_1, \dots dx_{k_j-n}) \mu^n(dz_1, \dots, dz_n)

always exist for 1\le n\le k_i, 1\le i,j\le d.
Therefore, by applying the Lebesgue dominated convergence theorem, we obtain that \Sigma^{(l)}(i,j)\to \Sigma(i,j) and \mathbf{E}(F_{i,l})\to\mathbf{E}(F_{i}) for l\to\infty. Hence,

\sqrt{C(\Sigma^{(l)})^{-1}}\left(F^{(l)}-\mathbf{E}(F^{(l)})\right)\to \sqrt{C\Sigma^{-1}}\left(F-\mathbf{E}(F)\right)

almost surely for l\to\infty.
Note that, the almost sure convergence implies the convergence in the probabilistic distance \Delta and |M^{(l)}_{n,m}(i,j)|\le |M_{n,m}(i,j)| , where M^{(l)}_{n,m}(i,j) is defined when we replace \phi_{i},\phi_{j} by \phi_{i}^{(l)},\phi_{j}^{(l)} in (7). Therefore, by using Theorem 4.1 and applying the triangular inequality, we conclude that

Theorem 4.2. Assume that F=(F_1,\ldots,F_d)\subset L^2(P_N) is a vector of U-statistics in the form (6) such that

\sum_{({z}_1,\dots,{z}_{k_i})\in S_{k_i}(N)} |\phi_i({z}_1,\dots,{z}_{k_i})|\in L^2(P_N).


\begin{array}{lll}\displaystyle\Delta\left(\sqrt{C\Sigma^{-1}}\left(F-\mathbf{E}(F)\right),X\right) \leq \\ \displaystyle  \cfrac{ \sqrt{2\pi}}{8}d^{2} k^{7/2} \|\sqrt{C\Sigma^{-1}}\|^3 \|C^{-1}\|^{3/2} \|C\| ({\rm trace}(\Sigma))^{1/2} \sqrt{\sum_{i=1}^d\sum_{n=1}^k M_{n,n}(i,i)}\\ \displaystyle +  k^2\|C\Sigma^{-1}\|_{F} \|C^{-1}\| \|C\|^{1/2}\sqrt{\sum_{i,j=1}^{d}\sum_{n,m=1}^{k} M_{n,m}(i,j) }, \end{array}

where k=\max\{k_i, 1\le i\le,d\} and M_{n,m}(i,j), 1\le i,j\le d, 1\le n,m\le k are defined in (7).

Corollary 4.3. Assume that \{F^{(l)}\}_{l\ge 0} is a sequence of vectors of U-statistics, which are defined as in Theorem 4.2, such that

\max_{1\le i,j\le d, 1\le n,m\le k}M_{n,m}^{(l)}(i,j)\to 0

for l\to\infty, then the law of \sqrt{C(\Sigma^{(l)})^{-1}}\left(F^{(l)}-\mathbf{E}(F^{(l)})\right) converges to the multivariate Gaussian law  \mathcal{N}_d(0,C).


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Đạo hàm Malliavin

July 12, 2010 Leave a comment

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<p,q<\infty\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

July 12, 2010 Leave a comment

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.