Central limit theorems and bootstrap in high dimensions

Victor Chernozhukov, Denis Chetverikov () and Kengo Kato
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Denis Chetverikov: Institute for Fiscal Studies and UCLA
Kengo Kato: Institute for Fiscal Studies

No CWP49/14, CeMMAP working papers from Centre for Microdata Methods and Practice, Institute for Fiscal Studies

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 Xi is in 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-> 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.

Date: 2014-12-31
New Economics Papers: this item is included in nep-ecm
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Related works:
Working Paper: Central limit theorems and bootstrap in high dimensions (2016)
Working Paper: Central limit theorems and bootstrap in high dimensions (2016)
Working Paper: Central limit theorems and bootstrap in high dimensions (2014)
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