Refined Berry-Esseen bounds under local dependence

TL;DR

This paper derives improved Berry-Esseen bounds for sums of locally dependent random variables using Stein's method and new concentration inequalities, applicable to various complex dependence structures.

Contribution

It introduces a novel class of concentration inequalities and extends Berry-Esseen bounds to broader local dependence models with optimal convergence rates.

Findings

Sharper bounds for graph dependency sums

Enhanced bounds for distributed U-statistics

Optimal convergence rates under general dependence

Abstract

In this paper, we establish Berry--Esseen bounds for both self-normalized and non-self-normalized sums of locally dependent random variables. The proofs are based on Stein's method together with a concentration inequality approach. We develop a new class of concentration inequalities that extend classical results and achieve optimal convergence rates under more general dependence structures. As applications, we apply our main results to derive sharper Berry--Esseen bounds for graph dependency, distributed $U$-statistics, constrained $U$-statistics, and decorated injective homomorphism sums.