VECTOR-VALUED STATISTICS OF BINOMIAL PROCESSES: BERRY-ESSEEN BOUNDS IN THE CONVEX DISTANCE

• 成果类型: Article
• 署名作者: Kasprzak, Mikolaj J.; Peccati, Giovanni
• 署名单位: University of Luxembourg
• 刊物名称: ANNALS OF APPLIED PROBABILITY
• ISSN/ISSBN: 1050-5164
• DOI: 10.1214/22-AAP1897
• 发表日期: 2023
• 页码: 3449-3492
• 关键词: multivariate normal approximation; central limit-theorems; exchangeable pairs; steins method; CONVERGENCE; rates; clt; functionals

摘要

We study the discrepancy between the distribution of a vector-valued functional of i.i.d. random elements and that of a Gaussian vector. Our main contribution is an explicit bound on the convex distance between the two distributions, holding in every dimension. Such a finding constitutes a substantial extension of the one-dimensional bounds deduced in Chatterjee (Ann. Probab. 36 (2008) 1584-1610) and Lachieze-Rey and Peccati (Ann. Appl. Probab. 27 (2017) 1992-2031), as well as of the multidimensional bounds for smooth test functions and indicators of rectangles derived, respectively, in Dung (Acta Math. Hungar. 158 (2019) 173-201), and Fang and Koike (Ann. Appl. Probab. 31 (2021) 1660-1686). Our techniques involve the use of Stein's method, combined with a suitable adaptation of the recursive approach inaugurated by Schulte and Yukich (Electron. J. Probab. 24 (2019) 1-42): this yields rates of converge that have a presumably optimal dependence on the sample size. We develop several applications of a geometric nature, among which is a new collection of multidimensional quantitative limit theorems for the intrinsic volumes associated with coverage processes in Euclidean spaces.