Berry--Esseen bounds for generalized $U$ statistics

TL;DR

This paper derives optimal Berry--Esseen bounds for generalized U-statistics using Stein's method, providing precise convergence rates for normal approximations in graph-related applications.

Contribution

It introduces a new Berry--Esseen theorem for exchangeable pairs and applies it to improve convergence rate results for subgraph counts in random graphs.

Findings

Optimal convergence rate for normal approximation of subgraph counts

New Berry--Esseen theorem for exchangeable pairs

Enhanced understanding of U-statistics in graph theory

Abstract

In this paper, we establish optimal Berry--Esseen bounds for the generalized $U$-statistics. The proof is based on a new Berry--Esseen theorem for exchangeable pair approach by Stein's method under a general linearity condition setting. As applications, an optimal convergence rate of the normal approximation for subgraph counts in Erd\"os--R\'enyi graphs and graphon-random graph is obtained.