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 ∑n i=1 Xi ∈ A) where X1, . . . , Xn are independent random vectors in R and A is a hyperrectangle, or, more generally, a sparsely convex set, and show that the approximation error converges to zero even if p = pn → ∞ as n→∞ and p n; in particular, p can be as large as O(e 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 Xi. 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.