On the Gaussian approximation of vector-valued multiple integrals

JOURNAL OF MULTIVARIATE ANALYSIS

卷:102

Article

Noreddine, Salim²  Nourdin, Ivan¹ 

[1] Univ Nancy 1, Inst Elie Cartan, F-54506 Vandoeuvre Les Nancy, France

[2] Univ Paris 06, Lab Probabil & Modeles Aleatoires, F-75252 Paris 5, France

关键词: Central limit theorem;    Cumulants;    Malliavin calculus;    Multiple integrals;    Ornstein-Uhlenbeck semigroup;   

DOI  :  10.1016/j.jmva.2011.02.001

来源: Elsevier

【 摘 要 】

By combining the findings of two recent, seminal papers by Nualart, Peccati and Tudor, we get that the convergence in law of any sequence of vector-valued multiple integrals F-n towards a centered Gaussian random vector N, with given covariance matrix C, is reduced to just the convergence of: (i) the fourth cumulant of each component of F-n to zero: (ii) the covariance matrix of F-n to C. The aim of this paper is to understand more deeply this somewhat surprising phenomenon. To reach this goal, we offer two results of a different nature. The first one is an explicit bound for d(F, N) in terms of the fourth cumulants of the components of F, when F is a R-d-valued random vector whose components are multiple integrals of possibly different orders, N is the Gaussian counterpart of F (that is, a Gaussian centered vector sharing the same covariance with F) and d stands for the Wasserstein distance. The second one is a new expression for the cumulants of F as above, from which it is easy to derive yet another proof of the previously quoted result by Nualart, Peccati and Tudor. (C) 2011 Elsevier Inc. All rights reserved.

【 授权许可】

Free