Central limit theorems and bootstrap in high dimensions

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 $\mathrm{P}(n^{-1/2}\sum_{i=1}^{n}X_{i}\in A)$ 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.