Stein's method on Wiener chaos

• 成果类型: Article
• 署名作者: Nourdin, Ivan; Peccati, Giovanni
• 署名单位: Universite Paris Cite; Sorbonne Universite; Sorbonne Universite
• 刊物名称: PROBABILITY THEORY AND RELATED FIELDS
• ISSN/ISSBN: 0178-8051
• DOI: 10.1007/s00440-008-0162-x
• 发表日期: 2009
• 页码: 75-118
• 关键词: CENTRAL LIMIT-THEOREMS; STOCHASTIC INTEGRALS; functionals; CONVERGENCE; approximations; inequalities

摘要

We combine Malliavin calculus with Stein's method, in order to derive explicit bounds in the Gaussian and Gamma approximations of random variables in a fixed Wiener chaos of a general Gaussian process. Our approach generalizes, refines and unifies the central and non-central limit theorems for multiple Wiener-It integrals recently proved (in several papers, from 2005 to 2007) by Nourdin, Nualart, Ortiz-Latorre, Peccati and Tudor. We apply our techniques to prove Berry-Esseen bounds in the Breuer-Major CLT for subordinated functionals of fractional Brownian motion. By using the well-known Mehler's formula for Ornstein-Uhlenbeck semi-groups, we also recover a technical result recently proved by Chatterjee, concerning the Gaussian approximation of functionals of finite-dimensional Gaussian vectors.