Dual Induction CLT for High-dimensional m-dependent Data

TL;DR

This paper establishes sharp high-dimensional Berry--Esseen bounds for sums of m-dependent vectors, achieving near-optimal rates with minimal assumptions, and introduces a novel inductive approach linking anti-concentration and Berry--Esseen bounds.

Contribution

It provides the first sharp high-dimensional bounds for m-dependent data with poly-logarithmic dimension dependence and develops a new inductive method inspired by classical techniques.

Findings

Achieves near-optimal sample complexity of m^{(q-1)/(q-2)}/√n.

Provides sharp bounds under minimal assumptions like finite third moments.

Introduces a novel inductive relationship between anti-concentration and Berry--Esseen bounds.

Abstract

We derive novel and sharp high-dimensional Berry--Esseen bounds for the sum of $m$-dependent random vectors over the class of hyper-rectangles exhibiting only a poly-logarithmic dependence in the dimension. Our results hold under minimal assumptions, such as non-degenerate covariances and finite third moments, and exhibit an optimal sample complexity of order $m^{(q-1)/(q-2)}/\sqrt{n}$. Aside from logarithmic terms, the resulting rates match the optimal rates established in the univariate case. When specialized to the sums of independent non-degenerate random vectors, our results produce sharp and, in some cases, optimal rates under the weakest possible conditions. We develop a novel inductive relationship between anti-concentration inequalities and Berry--Esseen bounds inspired by the classical Lindeberg swapping method and the concentration inequality approach for dependent data that may be of independent interest.