RATES IN THE CENTRAL LIMIT THEOREM FOR RANDOM PROJECTIONS OF MARTINGALES

• 成果类型: Article
• 署名作者: Dedecker, Jerome; Merlevede, Florence; Peligrad, Magda
• 署名单位: Universite Paris Cite; Centre National de la Recherche Scientifique (CNRS); CNRS - National Institute for Mathematical Sciences (INSMI); IMT - Institut Mines-Telecom; Institut Polytechnique de Paris; Telecom SudParis; Universite Paris-Est-Creteil-Val-de-Marne (UPEC); Universite Gustave-Eiffel; University System of Ohio; University of Cincinnati
• 刊物名称: ANNALS OF APPLIED PROBABILITY
• ISSN/ISSBN: 1050-5164
• DOI: 10.1214/24-AAP2121
• 发表日期: 2025
• 页码: 564-589
• 关键词: convergence; sums

摘要

In this paper, we consider partial sums of martingale differences weighted by random variables drawn uniformly on the sphere, and globally independent of the martingale differences. Combining Lindeberg's method and a series of arguments due to Bobkov, Chistyakov and G & ouml;tze, we show that the Kolmogorov distance between the distribution of these weighted sums and the limiting Gaussian is super-fast of order (log n)2/n, under conditions allowing us to control the higher-order conditional moments of the martingale differences. We also show that the same rate is achieved if we consider a quantity very close to these weighted sums, and give an application of this result to the least squares estimator of the slope in the linear model with Gaussian design.