LARGE-DIMENSIONAL CENTRAL LIMIT THEOREM WITHFOURTH-MOMENT ERROR BOUNDS ON CONVEX SETS AND BALLS

• 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: 10.1214/23-AAP2014
• 发表日期: 2024
• 页码: 2065-2106
• 关键词: multivariate normal approximation; quadratic-forms; bootstrap; CONVERGENCE; inequalities; inference; rates; clt

摘要

We prove the large-dimensional Gaussian approximation of a sum ofnindependent random vectors inRdtogether with fourth-moment error boundson convex sets and Euclidean balls. Our bounds have near-optimal depen-dence onnand, compared with classical third-moment bounds, can achieveimproved dependence on the dimensiond. For centered balls, we obtain anadditional error bound that has a sub-optimal dependence onn, but recoversthe known result of the validity of the Gaussian approximation if and onlyifd=o(n). We discuss an application to the bootstrap. We prove our mainresults using Stein's method.