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 X_i is in A, where X₁,..., X_nare independent random vectors in R^p and A is a rectangle, or, more generally, a sparsely convex set, and show that the approximation error converges to zero even if p=p_n-> 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 X_i. 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.