Central Limit Theorems and Bootstrap in High Dimensions

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 Pr(n−1/2∑ni=1Xi∈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→∞ and p≫n; in particular, p can be as large as O(eCnc) 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. All of the bounds on approximation errors are non-asymptotic. The proofs rely on an effective use of Slepian-Stein methods and some new Gaussian comparison inequalities.
These results have already found many useful applications in modern high-dimensional statistics and econometrics.

(This talk is based on joint work with V. Chernozhukov and K. Kato.)