Non-uniform Berry--Esseen bounds for Gaussian, Poisson and Rademacher processes

Abstract

In this paper we obtain non-uniform Berry-Esseen bounds for normal approximations by the Malliavin-Stein method. The techniques rely on a detailed analysis of the solutions of Stein's equations and will be applied to functionals of a Gaussian process like multiple Wiener-It\^o integrals, to Poisson functionals as well as to the Rademacher chaos expansion. Second-order Poincar\'e inequalities for normal approximation of these functionals are connected with non-uniform bounds as well. As applications, elements living inside a fixed Wiener chaos associated with an isonormal Gaussian process, like the discretized version of the quadratic variation of a fractional Brownian motion, are considered. Moreover we consider subgraph counts in random geometric graphs as an example of Poisson $U$-statistics, as well as subgraph counts in the Erd\H{o}s-R\'enyi random graph and infinite weighted 2-runs as examples of functionals of Rademacher variables.