Berry-Esseen Bounds of Normal and Non-normal Approximation for Unbounded Exchangeable Pairs

TL;DR

This paper develops new Berry-Esseen bounds for normal and non-normal approximation using Stein's method with exchangeable pairs, removing the bounded difference restriction and achieving optimal convergence rates in various models.

Contribution

It introduces a novel Berry-Esseen bound for exchangeable pairs without the bounded difference assumption, enhancing approximation accuracy.

Findings

Established a new Berry-Esseen bound for unbounded exchangeable pairs.

Achieved optimal convergence rates in multiple models including Curie-Weiss and graph models.

Extended Stein's method applicability to broader classes of exchangeable pairs.

Abstract

An exchangeable pair approach is commonly taken in the normal and non-normal approximation using Stein's method. It has been successfully used to identify the limiting distribution and provide an error of approximation. However, when the difference of the exchangeable pair is not bounded by a small deterministic constant, the error bound is often not optimal. In this paper, using the exchangeable pair approach of Stein's method, a new Berry-Esseen bound for an arbitrary random variable is established without a bound on the difference of the exchangeable pair. An optimal convergence rate for normal and non-normal approximation is achieved when the result is applied to various examples including the quadratic forms, general Curie-Weiss model, mean field Heisenberg model and colored graph model.