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Preprints, Working Papers, ... Year : 2008

Stein's method on Wiener chaos

Abstract

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. We also prove results concerning random variables admitting a possibly infinite Wiener chaotic decomposition. Our approach generalizes, refines and unifies the central and non-central limit theorems for multiple Wiener-Itô 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 semigroups, we also recover a result recently proved by Chatterjee, in the context of limit theorems for linear statistics of eigenvalues of random matrices.
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Dates and versions

hal-00199024 , version 1 (18-12-2007)
hal-00199024 , version 2 (27-12-2007)
hal-00199024 , version 3 (25-01-2008)
hal-00199024 , version 4 (03-02-2008)
hal-00199024 , version 5 (10-05-2008)

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Ivan Nourdin, Giovanni Peccati. Stein's method on Wiener chaos. 2008. ⟨hal-00199024v3⟩
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