Refined Berry-Esseen bounds under local dependence

TL;DR

This work derives refined Berry-Esseen bounds for normal approximation in the presence of local dependence under LD1 and LD2, employing Stein's method coupled with new generalized concentration inequalities. It provides distinct BE bounds for non-self-normalized sums $W_1=S/\sigma$ and self-normalized sums $W_2=S/V$, with rates expressed in terms of neighborhood sizes, moments, and the dependence parameters $\kappa$ and $\tau$, augmented by a self-normalization factor $\lambda$. The authors apply these bounds to diverse settings—graph dependency, distributed $U$-statistics, constrained $U$-statistics for $m$-dependent sequences, and decorated injective homomorphism sums—obtaining sharper convergence rates than prior results and accommodating broader dependence structures. Collectively, the results advance practical normal approximation in complex dependent data, informing inference in networks, distributed computing, and pattern-matching problems where local dependence is intrinsic.

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.

Theorems & Definitions (42)

• Theorem 2.1
• Theorem 2.2
• Remark 2.3
• Theorem 3.1
• Remark 3.2
• Remark 3.3
• Theorem 3.4
• Remark 3.5
• Remark 3.6
• Example 3.7