HIGH-DIMENSIONAL CENTRAL LIMIT THEOREMS BY STEIN'S METHOD

• 成果类型: Article
• 署名作者: Fang, Xiao; Koike, Yuta
• 署名单位: Chinese University of Hong Kong; University of Tokyo
• 刊物名称: ANNALS OF APPLIED PROBABILITY
• ISSN/ISSBN: 1050-5164
• DOI: 10.1214/20-AAP1629
• 发表日期: 2021
• 页码: 1660-1686
• 关键词: multivariate normal approximation; Gaussian Approximation; U-statistics; explicit rates; clt; CONVERGENCE; bootstrap; Kernels; expansions; maxima

摘要

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 Gotze (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 log n factor. We also discuss an application to multiple Wiener-Ito integrals.