BERRY-ESSEEN BOUNDS OF NORMAL AND NONNORMAL APPROXIMATION FOR UNBOUNDED EXCHANGEABLE PAIRS

• 成果类型: Article
• 署名作者: Shao, Qi-Man; Zhang, Zhuo-Song
• 署名单位: Chinese University of Hong Kong
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• DOI: 10.1214/18-AOP1255
• 发表日期: 2019
• 页码: 61-108
• 关键词: multivariate normal approximation; steins method; clt

摘要

An exchangeable pair approach is commonly taken in the normal and nonnormal approximation using Stein's method. It has been successfully used to identify the limiting distribution and provide an error of approximation. However, when the difference of the exchangeable pair is not bounded by a small deterministic constant, the error bound is often not optimal. In this paper, using the exchangeable pair approach of Stein's method, a new Berry-Esseen bound for an arbitrary random variable is established without a bound on the difference of the exchangeable pair. An optimal convergence rate for normal and nonnormal approximation is achieved when the result is applied to various examples including the quadratic forms, general Curie-Weiss model, mean field Heisenberg model and colored graph model.