The Berry-Esseen Bound for High-dimensional Self-normalized Sums

TL;DR

This paper establishes a new Berry-Esseen bound for the Gaussian approximation of high-dimensional self-normalized sums, advancing the understanding of their distributional behavior under weak moment conditions.

Contribution

It provides the first explicit Berry-Esseen bound for high-dimensional self-normalized sums in the CLT context, with bounds depending on sample size and dimension.

Findings

Bound scales as log^{5/4}(d)/n^{1/8} under finite third absolute moment

Bound tends to zero if log(d)=o(n^{1/10})

Represents a significant advancement in high-dimensional CLT literature

Abstract

This manuscript studies the Gaussian approximation of the coordinate-wise maximum of self-normalized statistics in high-dimensional settings. We derive an explicit Berry-Esseen bound under weak assumptions on the absolute moments. When the third absolute moment is finite, our bound scales as $\log^{5/4}(d)/n^{1/8}$ where $n$ is the sample size and $d$ is the dimension. Hence, our bound tends to zero as long as $\log(d)=o(n^{1/10})$. Our results on self-normalized statistics represent substantial advancements, as such a bound has not been previously available in the high-dimensional central limit theorem (CLT) literature.