A Berry-Esseen bound with applications to vertex degree counts in the Erd\H{o}s-R\'{e}nyi random graph

TL;DR

This paper develops a Berry-Esseen bound using Stein's method that applies to normal approximation without bounded coupling restrictions, and demonstrates its use in analyzing vertex degree counts in Erdős-Rényi graphs.

Contribution

It introduces a new Berry-Esseen bound via Stein's method that relaxes previous bounded coupling constraints and applies it to degree counts in Erdős-Rényi graphs.

Findings

Established a Berry-Esseen bound without bounded coupling restrictions.

Applied the bound to vertex degree counts in Erdős-Rényi graphs.

Demonstrated the effectiveness of the method in a random graph context.

Abstract

Applying Stein's method, an inductive technique and size bias coupling yields a Berry-Esseen theorem for normal approximation without the usual restriction that the coupling be bounded. The theorem is applied to counting the number of vertices in the Erdos-Renyi random graph of a given degree.