CENTRAL LIMIT THEOREMS AND BOOTSTRAP IN HIGH DIMENSIONS

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 P(n(-1/2) Sigma(n)(i=1) X-i is an element of A) where X-1,...,X-n are independent random vectors in 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) -> infinity as n -> infinity and p >> n; in particular, p can be as large as O(e(Cnc)) 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.