Large-dimensional Central Limit Theorem with Fourth-moment Error Bounds on Convex Sets and Balls

TL;DR

This paper establishes a large-dimensional Gaussian approximation for sums of independent vectors with fourth-moment error bounds on convex sets and balls, improving classical bounds and discussing bootstrap applications.

Contribution

It provides near-optimal fourth-moment bounds for Gaussian approximation in high dimensions, with improved dependence on dimension and sample size, using Stein's method.

Findings

Bounds have near-optimal dependence on n

Improved dependence on the dimension d

Valid Gaussian approximation if and only if d=o(n)

Abstract

We prove the large-dimensional Gaussian approximation of a sum of $n$ independent random vectors in $\mathbb{R}^d$ together with fourth-moment error bounds on convex sets and Euclidean balls. We show that compared with classical third-moment bounds, our bounds have near-optimal dependence on $n$ and can achieve improved dependence on the dimension $d$. For centered balls, we obtain an additional error bound that has a sub-optimal dependence on $n$, but recovers the known result of the validity of the Gaussian approximation if and only if $d=o(n)$. We discuss an application to the bootstrap. We prove our main results using Stein's method.