Open Access
July 2017 Central limit theorems and bootstrap in high dimensions
Victor Chernozhukov, Denis Chetverikov, Kengo Kato
Ann. Probab. 45(4): 2309-2352 (July 2017). DOI: 10.1214/16-AOP1113


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 $\mathrm{P}(n^{-1/2}\sum_{i=1}^{n}X_{i}\in A)$ where $X_{1},\dots,X_{n}$ are independent random vectors in $\mathbb{R}^{p}$ and $A$ is a hyperrectangle, or more generally, a sparsely convex set, and show that the approximation error converges to zero even if $p=p_{n}\to\infty$ as $n\to\infty$ and $p\gg n$; in particular, $p$ can be as large as $O(e^{Cn^{c}})$ for some constants $c,C>0$. The result holds uniformly over all hyperrectangles, or more generally, sparsely convex sets, and does not require any restriction on the correlation structure among coordinates of $X_{i}$. 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.


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Victor Chernozhukov. Denis Chetverikov. Kengo Kato. "Central limit theorems and bootstrap in high dimensions." Ann. Probab. 45 (4) 2309 - 2352, July 2017.


Received: 1 April 2015; Revised: 1 March 2016; Published: July 2017
First available in Project Euclid: 11 August 2017

zbMATH: 1377.60040
MathSciNet: MR3693963
Digital Object Identifier: 10.1214/16-AOP1113

Primary: 60F05 , 62E17

Keywords: bootstrap limit theorems , central limit theorem , high dimensions , hyperrectangles , sparsely convex sets

Rights: Copyright © 2017 Institute of Mathematical Statistics

Vol.45 • No. 4 • July 2017
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