Berry-Esseen-type estimates for random variables with a sparse dependency graph

TL;DR

This paper derives improved Berry-Esseen bounds for sums of dependent random variables with sparse dependency graphs, using Fourier analysis, and establishes a CLT under bounded moment conditions.

Contribution

It introduces sharper Berry-Esseen bounds for dependent variables with large dependency graph degrees, advancing the understanding of CLTs in sparse dependency settings.

Findings

Enhanced bounds for sums with dependency graphs

Central Limit Theorem for variables with bounded moments

Fourier transform approach improves existing estimates

Abstract

We obtain Berry-Esseen-type bounds for the sum of random variables with a dependency graph and uniformly bounded moments of order $\delta \in (2,\infty]$ using a Fourier transform approach. Our bounds improve the state-of-the-art in the regime where the degree of the dependency graph is large. As a Corollary of our results, we obtain a Central Limit Theorem for random variables with a sparse dependency graph that are uniformly bounded in $L^{\delta}$ for some $\delta\in(2,\infty]$.