High-dimensional Berry-Esseen Bound for $m$-Dependent Random Samples

TL;DR

This paper establishes a finite sample Berry-Esseen bound for high-dimensional, m-dependent random vectors, extending classical results to dependent data with minimal assumptions.

Contribution

It provides a novel $(n/m)^{-1/2}$-rate Berry-Esseen bound for high-dimensional dependent samples under minimal conditions, using innovative inductive proof techniques.

Findings

Achieves tight bounds for dependent high-dimensional data.

Extends Berry-Esseen bounds to m-dependent vectors.

Uses inductive methods inspired by Chen, Shao, Kuchibhotla, and Rinaldo.

Abstract

In this work, we provide a $(n/m)^{-1/2}$-rate finite sample Berry-Esseen bound for $m$-dependent high-dimensional random vectors over the class of hyper-rectangles. This bound imposes minimal assumptions on the random vectors such as nondegenerate covariances and finite third moments. The proof uses inductive relationships between anti-concentration inequalities and Berry--Esseen bounds, which are inspired by the telescoping method of Chen and Shao (2004) and the recursion method of Kuchibhotla and Rinaldo (2020). Performing a dual induction based on the relationships, we obtain tight Berry-Esseen bounds for dependent samples.