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August 2021 High-dimensional central limit theorems by Stein’s method
Xiao Fang, Yuta Koike
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Ann. Appl. Probab. 31(4): 1660-1686 (August 2021). DOI: 10.1214/20-AAP1629

Abstract

We obtain explicit error bounds for the d-dimensional normal approximation on hyperrectangles for a random vector that has a Stein kernel, or admits an exchangeable pair coupling, or is a nonlinear statistic of independent random variables or a sum of n locally dependent random vectors. We assume the approximating normal distribution has a nonsingular covariance matrix. The error bounds vanish even when the dimension d is much larger than the sample size n. We prove our main results using the approach of Götze (1991) in Stein’s method, together with modifications of an estimate of Anderson, Hall and Titterington (1998) and a smoothing inequality of Bhattacharya and Rao (1976). For sums of n independent and identically distributed isotropic random vectors having a log-concave density, we obtain an error bound that is optimal up to a logn factor. We also discuss an application to multiple Wiener–Itô integrals.

Funding Statement

Fang X. was partially supported by Hong Kong RGC ECS 24301617 and GRF 14302418 and 14304917, a CUHK direct grant and a CUHK start-up grant. Koike Y. was partially supported by JST CREST Grant Number JPMJCR14D7 and JSPS KAKENHI Grant Numbers JP17H01100, JP18H00836, JP19K13668.

Acknowledgments

We thank the two anonymous referees for their careful reading of the manuscript and for their valuable suggestions which led to many improvements. We also thank Victor Chernozhukov for useful comments on the proof of Lemma 2.2.

Citation

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Xiao Fang. Yuta Koike. "High-dimensional central limit theorems by Stein’s method." Ann. Appl. Probab. 31 (4) 1660 - 1686, August 2021. https://doi.org/10.1214/20-AAP1629

Information

Received: 1 January 2020; Revised: 1 September 2020; Published: August 2021
First available in Project Euclid: 15 September 2021

Digital Object Identifier: 10.1214/20-AAP1629

Subjects:
Primary: 60F05 , 62E17

Keywords: central limit theorem , Exchangeable pairs , high dimensions , local dependence , multiple Wiener–Itô integrals , nonlinear statistic , Stein kernel , Stein’s method

Rights: Copyright © 2021 Institute of Mathematical Statistics

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Vol.31 • No. 4 • August 2021
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