June 2023 Nearly optimal central limit theorem and bootstrap approximations in high dimensions
Victor Chernozhukov, Denis Chetverikov, Yuta Koike
Author Affiliations +
Ann. Appl. Probab. 33(3): 2374-2425 (June 2023). DOI: 10.1214/22-AAP1870


In this paper, we derive new, nearly optimal bounds for the Gaussian approximation to scaled averages of n independent high-dimensional centered random vectors X1,,Xn over the class of rectangles in the case when the covariance matrix of the scaled average is nondegenerate. In the case of bounded Xi’s, the implied bound for the Kolmogorov distance between the distribution of the scaled average and the Gaussian vector takes the form


where d is the dimension of the vectors and Bn is a uniform envelope constant on components of Xi’s. This bound is sharp in terms of d and Bn, and is nearly (up to logn) sharp in terms of the sample size n. In addition, we show that similar bounds hold for the multiplier and empirical bootstrap approximations. Moreover, we establish bounds that allow for unbounded Xi’s, formulated solely in terms of moments of Xi’s. Finally, we demonstrate that the bounds can be further improved in some special smooth and moment-constrained cases.

Funding Statement

Y. Koike was partly supported by JST CREST Grant Number JPMJCR2115 and JSPS KAKENHI Grant Number JP19K13668.


We would like to thank two anonymous referees for their constructive comments. We are also grateful to Nilanjan Chakraborty, Xiaohong Chen, Xiao Fang and Kengo Kato for helpful discussions.


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Victor Chernozhukov. Denis Chetverikov. Yuta Koike. "Nearly optimal central limit theorem and bootstrap approximations in high dimensions." Ann. Appl. Probab. 33 (3) 2374 - 2425, June 2023. https://doi.org/10.1214/22-AAP1870


Received: 1 May 2021; Revised: 1 June 2022; Published: June 2023
First available in Project Euclid: 2 May 2023

MathSciNet: MR4583674
zbMATH: 07692321
Digital Object Identifier: 10.1214/22-AAP1870

Primary: 60F05 , 62E17

Keywords: Berry–Esseen bound , bootstrap limit theorems , central limit theorem , high dimensions , smoothing inequalities

Rights: Copyright © 2023 Institute of Mathematical Statistics


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Vol.33 • No. 3 • June 2023
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