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 factor. We also discuss an application to multiple Wiener–Itô integrals.
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.
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.
"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