BERRY-ESSEEN TYPE BOUNDS FOR THE LEFT RANDOM WALK ON GLd(R) UNDER POLYNOMIAL MOMENT CONDITIONS

• 成果类型: Article
• 署名作者: Cuny, C.; Dedecker, J.; Merlevede, F.; Peligrad, M.
• 署名单位: Universite de Bretagne Occidentale; Universite Paris Cite; Universite Gustave-Eiffel; Universite Paris-Est-Creteil-Val-de-Marne (UPEC); University System of Ohio; University of Cincinnati
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• DOI: 10.1214/22-AOP1602
• 发表日期: 2023
• 页码: 495-523
• 关键词: CENTRAL-LIMIT-THEOREM; CONVERGENCE; inequalities; rates

摘要

Let A(n) = epsilon(n) center dot center dot center dot epsilon(1), where (epsilon n)n >= 1 is a sequence of independent random matrices, taking values in GL(d)(R), d >= 2, with common distribution mu. In this paper, under standard assumptions on mu (strong irreducibility and prox-imality) we prove Berry-Esseen type theorems for log(||A(n)||) when mu has a polynomial moment. More precisely, we get the rate ((logn)/n)(q/2-1), when mu has a moment of order q is an element of[2, 3] and the rate 1/root n when mu has a moment of order 4, which significantly improves earlier results in this setting.