Notes on the dimension dependence in high-dimensional central limit theorems for hyperrectangles

TL;DR

This paper establishes that high-dimensional CLTs for hyperrectangles hold under certain growth conditions on dimension and sample size, providing a theoretical basis for high-dimensional inference.

Contribution

It improves existing dimension growth conditions for Gaussian approximation in high-dimensional CLTs and extends results to bootstrap methods.

Findings

CLT approximation holds if $( ext{log } d)^5/n o 0$

Improved condition to $( ext{log } d)^3/n o 0$ with common factors

Bootstrap approximation results are also established

Abstract

Let $X_1,\dots,X_n$ be independent centered random vectors in $\mathbb{R}^d$. This paper shows that, even when $d$ may grow with $n$, the probability $P(n^{-1/2}\sum_{i=1}^nX_i\in A)$ can be approximated by its Gaussian analog uniformly in hyperrectangles $A$ in $\mathbb{R}^d$ as $n\to\infty$ under appropriate moment assumptions, as long as $(\log d)^5/n\to0$. This improves a result of Chernozhukov, Chetverikov & Kato [Ann. Probab. 45 (2017) 2309-2353] in terms of the dimension growth condition. When $n^{-1/2}\sum_{i=1}^nX_i$ has a common factor across the components, this condition can be further improved to $(\log d)^3/n\to0$. The corresponding bootstrap approximation results are also developed. These results serve as a theoretical foundation of simultaneous inference for high-dimensional models.