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^-1/2₈=₁^nX₈ A) where X₁, , X₍ 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₍ as n and p 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 X₈. 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.