Central Limit Theorems and Bootstrap in High Dimensions

TL;DR

This paper establishes Gaussian and bootstrap approximation theorems for high-dimensional sums of random vectors hitting hyperrectangles and sparsely convex sets, with errors diminishing even when the dimension grows exponentially with the sample size.

Contribution

It extends CLT and bootstrap results to high-dimensional settings with minimal restrictions on correlation structures and for very large dimensions.

Findings

Gaussian approximation error converges to zero as dimension grows.

Bootstrap methods are valid in ultra-high-dimensional regimes.

Results hold uniformly over hyperrectangles and sparsely convex sets.

Abstract

This paper derives central limit and bootstrap theorems for probabilities that sums of centered high-dimensional random vectors hit hyperrectangles and sparsely convex sets. Specifically, we derive Gaussian and bootstrap approximations for probabilities $\Pr(n^{-1/2}\sum_{i=1}^n X_i\in A)$ where $X_1,\dots,X_n$ are independent random vectors in $\mathbb{R}^p$ and $A$ is a hyperrectangle, or, more generally, a sparsely convex set, and show that the approximation error converges to zero even if $p=p_n\to \infty$ as $n \to \infty$ and $p \gg n$; in particular, $p$ can be as large as $O(e^{Cn^c})$ for some constants $c,C>0$. The result holds uniformly over all hyperrectangles, or more generally, sparsely convex sets, and does not require any restriction on the correlation structure among coordinates of $X_i$. Sparsely convex sets are sets that can be represented as intersections of many convex sets whose indicator functions depend only on a small subset of their arguments, with hyperrectangles being a special case.