STEIN'S METHOD FOR CONDITIONAL CENTRAL LIMIT THEOREM

• 成果类型: Article
• 署名作者: Dey, Partha s.; Terlov, G. R. I. G. O. R. Y.
• 署名单位: University of Illinois System; University of Illinois Urbana-Champaign
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• DOI: 10.1214/22-AOP1613
• 发表日期: 2023
• 页码: 723-773
• 关键词: multivariate normal approximation; poisson convergence; exchangeable pairs; variables; bounds; sums

摘要

In the seventies, Charles Stein revolutionized the way of proving the cen-tral limit theorem by introducing a method that utilizes a characterization equation for Gaussian distribution. In the last 50 years, much research has been done to adapt and strengthen this method to a variety of different set-tings and other limiting distributions. However, it has not been yet extended to study conditional convergences. In this article we develop a novel approach, using Stein's method for exchangeable pairs, to find a rate of convergence in the conditional central limit theorem of the form (X-n | Y-n= k), where (X-n, Y-n) are asymptotically jointly Gaussian, and extend this result to a mul-tivariate version. We apply our general result to several concrete examples, including pattern count in a random binary sequence and subgraph count in Erdos-Renyi random graph.