CENTRAL LIMIT THEOREM AND BOOTSTRAP APPROXIMATION IN HIGH DIMENSIONS: NEAR 1/√n RATES VIA IMPLICIT SMOOTHING

• 成果类型: Article
• 署名作者: Lopes, Miles E.
• 署名单位: University of California System; University of California Davis
• 刊物名称: ANNALS OF STATISTICS
• ISSN/ISSBN: 0090-5364
• DOI: 10.1214/22-AOS2184
• 发表日期: 2022
• 页码: 2492-2513
• 关键词: maxima; sums

摘要

Nonasymptotic bounds for Gaussian and bootstrap approximation have recently attracted significant interest in high-dimensional statistics. This paper studies Berry-Esseen bounds for such approximations with respect to the multivariate Kolmogorov distance, in the context of a sum of n random vectors that are p-dimensional and i.i.d. Up to now, a growing line of work has established bounds with mild logarithmic dependence on p. However, the problem of developing corresponding bounds with near n(-1/2) dependence on n has remained largely unresolved. Within the setting of random vectors that have sub-Gaussian or subexponential entries, this paper establishes bounds with near n(-1/2) dependence, for both Gaussian and bootstrap approximation. In addition, the proofs are considerably distinct from other recent approaches, and make use of an implicit smoothing operation in the Lindeberg interpolation.