Central limit theorems and bootstrap in high dimensions

In this paper, we derive central limit and bootstrap theorems for probabilities that centered high-dimensional vector sums hit rectangles and sparsely convex sets. Specifically, we derive Gaussian and bootstrap approximations for the probabilities that a root-n rescaled sample average of Xi is in A, where X1,..., Xn are independent random vectors in Rp and A is a rectangle, or, more generally, a sparsely convex set, and show that the approximation error converges to zero even if p=pn-> infinity 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 rectangles, or more generally, sparsely convex sets, and does not require any restrictions on the correlation among components of Xi. Sparsely convex sets are sets that can be represented as intersections of many convex sets whose indicator functions depend nontrivially only on a small subset of their arguments, with rectangles being a special case.