Stein’s method on Wiener chaos

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 multi...

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Bibliographic Details
Published in:Probability theory and related fields 2009-09, Vol.145 (1-2), p.75-118
Main Authors: Nourdin, Ivan, Peccati, Giovanni
Format: Article
Language:English
Subjects:
Approximation
Brownian motion
Calculus
Economics
Exact sciences and technology
Finance
Gaussian process
General topics
Insurance
Integrals
Limit theorems
Management
Mathematical and Computational Biology
Mathematical and Computational Physics
Mathematical functions
Mathematics
Mathematics and Statistics
Operations Research/Decision Theory
Probability
Probability and statistics
Probability Theory and Stochastic Processes
Quantitative Finance
Random variables
Sciences and techniques of general use
Statistics for Business
Stochastic processes
Studies
Theoretical
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Summary: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–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–Esséen 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 technical result recently proved by Chatterjee, concerning the Gaussian approximation of functionals of finite-dimensional Gaussian vectors.
ISSN:0178-8051
1432-2064
DOI:10.1007/s00440-008-0162-x