Stein's method and stochastic analysis of Rademacher functionals

TL;DR

This paper develops explicit bounds for Gaussian approximation of Rademacher functionals using Stein's method and discrete Malliavin calculus, with applications to CLTs for chaos, infinite 2-runs, and sparse multiple integrals.

Contribution

It introduces a novel approach combining Stein's method with chaos expansion for Rademacher functionals without classical exchangeable pairs.

Findings

Derived explicit Gaussian bounds for Rademacher functionals.

Applied methods to CLTs for fixed chaos and infinite 2-runs.

Provided an alternative proof and refinements for CLTs over sparse sets.

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.