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

TL;DR

This work tackles the high-dimensional CLT for the coordinate-wise maximum of self-normalized sums by establishing an explicit Berry-Esseen bound under mild moment conditions. It develops two Gaussian-approximation schemes, the Best Gaussian Approximation $\Delta_n$ and the Moment Matching $\Delta_n^X$, and shows that $\Delta_n$ decays at rate $\log^{5/4}(ed)\,n^{-1/8}$ when the third absolute moment is finite, vanishing as long as $\log d = o(n^{1/10})$. The analysis employs truncation, smoothing, and a Lindeberg-style swapping argument, with a detailed treatment of error terms and anti-concentration to connect the self-normalized sums to Gaussian limits in a high-dimensional setting. These results extend self-normalized CLT literature by providing finite-sample-type bounds in dimensionality that can grow with $n$, enabling valid inference and bootstrap-based methods in high-dimensional contexts.

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.

Theorems & Definitions (20)

• Theorem 1
• Corollary 1.1
• Remark 1: On the rate of convergence
• proof : Proof of \ref{['cor:berry-esseen']}
• Lemma 2
• proof : Proof of \ref{['lemma:truncation']}
• Lemma 3
• proof : Proof of \ref{['lemma:remainder1']}
• Lemma 4
• proof : Proof of \ref{['lemma:remainder2']}