Normal approximation and almost sure central limit theorem for non-symmetric Rademacher functionals

TL;DR

This paper advances the understanding of normal approximation and almost sure central limit theorems for Rademacher functionals by introducing a new chain rule and applying Malliavin-Stein methods.

Contribution

It provides a novel chain rule for Rademacher functionals and establishes almost sure CLTs using the Ibragimov-Lifshits criterion.

Findings

Improved chain rule for Rademacher functionals

Bound on Wasserstein distance for normal approximation

Almost sure CLT for Rademacher chaos

Abstract

In this work, we study the normal approximation and almost sure central limit theorems for some functionals of an independent sequence of Rademacher random variables. In particular, we provide a new chain rule that improves the one derived by Nourdin, Peccati and Reinert(2010) and then we deduce the bound on Wasserstein distance for normal approximation using the (discrete) Malliavin-Stein approach. Besides, we are able to give the almost sure central limit theorem for a sequence of random variables inside a fixed Rademacher chaos using the Ibragimov-Lifshits criterion.