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Stein’s method on Wiener chaos
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  • Published: 20 June 2008

Stein’s method on Wiener chaos

  • Ivan Nourdin1 &
  • Giovanni Peccati2 

Probability Theory and Related Fields volume 145, pages 75–118 (2009)Cite this article

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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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Author information

Authors and Affiliations

  1. Laboratoire de Probabilités et Modèles Aléatoires, Université Pierre et Marie Curie, Boîte courrier 188, 4 Place Jussieu, 75252, Paris Cedex 5, France

    Ivan Nourdin

  2. Laboratoire de Statistique Théorique et Appliquée, Université Pierre et Marie Curie, 8ème étage, bâtiment A, 175 rue du Chevaleret, 75013, Paris, France

    Giovanni Peccati

Authors
  1. Ivan Nourdin
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  2. Giovanni Peccati
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Correspondence to Ivan Nourdin.

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Cite this article

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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  • Received: 18 December 2007

  • Revised: 10 May 2008

  • Published: 20 June 2008

  • Issue Date: September 2009

  • DOI: https://doi.org/10.1007/s00440-008-0162-x

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Keywords

  • Berry–Esséen bounds
  • Breuer–Major CLT
  • Fractional Brownian motion
  • Gamma approximation
  • Malliavin calculus
  • Multiple stochastic integrals
  • Normal approximation
  • Stein’s method

Mathematics Subject Classification (2000)

  • 60F05
  • 60G15
  • 60H05
  • 60H07
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