## The Annals of Probability

- Ann. Probab.
- Volume 45, Number 4 (2017), 2309-2352.

### Central limit theorems and bootstrap in high dimensions

Victor Chernozhukov, Denis Chetverikov, and Kengo Kato

#### Abstract

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.

#### Article information

**Source**

Ann. Probab., Volume 45, Number 4 (2017), 2309-2352.

**Dates**

Received: April 2015

Revised: March 2016

First available in Project Euclid: 11 August 2017

**Permanent link to this document**

https://projecteuclid.org/euclid.aop/1502438428

**Digital Object Identifier**

doi:10.1214/16-AOP1113

**Mathematical Reviews number (MathSciNet)**

MR3693963

**Zentralblatt MATH identifier**

1377.60040

**Subjects**

Primary: 60F05: Central limit and other weak theorems 62E17: Approximations to distributions (nonasymptotic)

**Keywords**

Central limit theorem bootstrap limit theorems high dimensions hyperrectangles sparsely convex sets

#### Citation

Chernozhukov, Victor; Chetverikov, Denis; Kato, Kengo. Central limit theorems and bootstrap in high dimensions. Ann. Probab. 45 (2017), no. 4, 2309--2352. doi:10.1214/16-AOP1113. https://projecteuclid.org/euclid.aop/1502438428