Electronic Journal of Probability

Stein's Method and Stochastic Analysis of Rademacher Functionals

Gesine Reinert, Ivan Nourdin, and Giovanni Peccati

Full-text: Open access

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. As the bounds are given in terms of Malliavin operators, no coupling construction is required. When the functional depends only on the first d coordinates of the Rademacher sequence, a simple sufficient condition for convergence to a normal distribution is derived. For finite quadratic forms, we obtain necessary and sufficient conditions. Although our approach does not require the classical use of exchangeable pairs, when the functional depends only on the first d coordinates of the Rademacher sequence we employ chaos expansion in order to construct an explicit exchangeable pair vector; the elements of the vector relate to the summands in the chaos decomposition and satisfy a linearity condition for the conditional expectation. Among several examples, such as 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.

Article information

Source
Electron. J. Probab., Volume 15 (2010), paper no. 55, 1703-1742.

Dates
Accepted: 3 November 2010
First available in Project Euclid: 1 June 2016

Permanent link to this document
https://projecteuclid.org/euclid.ejp/1464819840

Digital Object Identifier
doi:10.1214/EJP.v15-823

Mathematical Reviews number (MathSciNet)
MR2735379

Zentralblatt MATH identifier
1225.60046

Subjects
Primary: 60F05: Central limit and other weak theorems
Secondary: 60F99: None of the above, but in this section 60G50: Sums of independent random variables; random walks 60H07: Stochastic calculus of variations and the Malliavin calculus

Keywords
Central Limit Theorems Discrete Malliavin operators Normal approximation Rademacher sequences Sparse sets Stein's method Walsh chaos

Rights
This work is licensed under aCreative Commons Attribution 3.0 License.

Citation

Reinert, Gesine; Nourdin, Ivan; Peccati, Giovanni. Stein's Method and Stochastic Analysis of Rademacher Functionals. Electron. J. Probab. 15 (2010), paper no. 55, 1703--1742. doi:10.1214/EJP.v15-823. https://projecteuclid.org/euclid.ejp/1464819840


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