Berry--Esseen bounds for generalized $U$ statistics

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.