Stein's method and stochastic analysis of Rademacher functionals

Abstract : We compute explicit bounds in the Gaussian approximation of functionals of infinite Rademacher sequences. Our tools involve Stein's method, as well as the use of appropriate discrete Malliavin operators. Although our approach does not require the classical use of exchangeable pairs, we employ a chaos expansion in order to construct an explicit exchangeable pair vector for any random variable which depends on a finite set of Rademacher variables. Among several examples, which include random variables which depend on infinitely many Rademacher variables, we provide three main applications: (i) to CLTs for multilinear forms belonging to a fixed chaos, (ii) to the Gaussian approximation of weighted infinite 2-runs, and (iii) to the computation of explicit bounds in CLTs for multiple integrals over sparse sets. This last application provides an alternate proof (and several refinements) of a recent result by Blei and Janson.
Type de document :
Pré-publication, Document de travail
35 pages + Appendix. New version: some inaccuracies in Sect. 6 corrected. 2009
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https://hal.archives-ouvertes.fr/hal-00331332
Contributeur : Ivan Nourdin <>
Soumis le : jeudi 21 mai 2009 - 11:54:12
Dernière modification le : lundi 29 mai 2017 - 14:25:26
Document(s) archivé(s) le : jeudi 23 septembre 2010 - 17:10:07

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  • HAL Id : hal-00331332, version 3
  • ARXIV : 0810.2890

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Ivan Nourdin, Giovanni Peccati, Gesine Reinert. Stein's method and stochastic analysis of Rademacher functionals. 35 pages + Appendix. New version: some inaccuracies in Sect. 6 corrected. 2009. <hal-00331332v3>

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