Resumen de Central limit theorems and bootstrap in high dimensions
Victor Chernozhukov, Denis Chetverikov, Kengo Kato
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∑ni=1Xi∈A)P(n−1/2∑i=1nXi∈A) where X1,…,XnX1,…,Xn are independent random vectors in RpRp and AA is a hyperrectangle, or more generally, a sparsely convex set, and show that the approximation error converges to zero even if p=pn→∞p=pn→∞ as n→∞n→∞ and p≫np≫n; in particular, pp can be as large as O(eCnc)O(eCnc) for some constants c,C>0c,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 XiXi. 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.
