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Central limit theorems and bootstrap in high dimensions

Authors: Victor Chernozhukov , Denis Chetverikov and Kengo Kato
Date: 26 August 2016
Type: cemmap Working Paper, CWP39/16
DOI: 10.1920/wp.cem.2016.3916

Abstract

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 Xis in A, where X1,..., Xnare independent random vectors in Rp and 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.

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Previous version:
Victor Chernozhukov, Denis Chetverikov and Kengo Kato December 2014, Central limit theorems and bootstrap in high dimensions, cemmap Working Paper, CWP49/14, IFS

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