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. 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.
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Nourdin, I., Peccati, G. Stein’s method on Wiener chaos. Probab. Theory Relat. Fields 145, 75–118 (2009). https://doi.org/10.1007/s00440-008-0162-x
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DOI: https://doi.org/10.1007/s00440-008-0162-x
Keywords
- Berry–Esséen bounds
- Breuer–Major CLT
- Fractional Brownian motion
- Gamma approximation
- Malliavin calculus
- Multiple stochastic integrals
- Normal approximation
- Stein’s method