NEARLY OPTIMAL CENTRAL LIMIT THEOREM AND BOOTSTRAP APPROXIMATIONS IN HIGH DIMENSIONS

• 成果类型: Article
• 署名作者: Chernozhukov, Victor; Chetverikov, Denis; Koike, Yuta
• 署名单位: Massachusetts Institute of Technology (MIT); Massachusetts Institute of Technology (MIT); University of California System; University of California Los Angeles; University of Tokyo; University of Tokyo
• 刊物名称: ANNALS OF APPLIED PROBABILITY
• ISSN/ISSBN: 1050-5164
• DOI: 10.1214/22-AAP1870
• 发表日期: 2023
• 页码: 2374-2425
• 关键词: multivariate normal approximation; Gaussian Approximation; steins method; sums; CONVERGENCE; dependence; deviation; moderate; suprema; maxima

摘要

In this paper, we derive new, nearly optimal bounds for the Gaussian ap-proximation 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 C(Bn2 log3 d/n)1/2log n, where d is the dimension of the vectors and Bn is a uniform envelope con-stant on components of Xi's. This bound is sharp in terms of d and Bn, and is nearly (up to log n) sharp in terms of the sample size n. In addition, we show that similar bounds hold for the multiplier and empirical bootstrap ap-proximations. 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.