Strengthening classical results on convergence rates in strong limit theorems

• 成果类型: Article
• 署名作者: Spataru, Aurel
• 署名单位: Romanian Academy; Institute of Mathematical Statistics & Applied Mathematics of Romanian Academy
• 刊物名称: PROBABILITY THEORY AND RELATED FIELDS
• ISSN/ISSBN: 0178-8051
• DOI: 10.1007/s00440-004-0404-5
• 发表日期: 2006
• 页码: 1-18
• 关键词: sums; LAW; probabilities; inequalities

摘要

Let X-1, X-2, . . . be i.i.d. random variables, and set S-n=X-1+ . . . +X-n. Several authors proved convergence of series of the type f(epsilon)=Sigma(n)c(n)P(|S-n| > epsilon a(n)), epsilon >alpha, under necessary and sufficient conditions. We show that under the same conditions, in fact integral(infinity)(delta) f(epsilon)d epsilon < infinity, delta > alpha, i.e. the finiteness of Sigma(n)c(n)P(|S-n| > epsilon a(n)), epsilon >alpha, is equivalent to the convergence of the double sum Sigma(k)Sigma(n)c(n)P(|S-n| > ka(n)). Two exceptional series required deriving necessary and sufficient conditions for E[sup(n)|S-n|(log(n))(eta)/n] < infinity, 0 <= eta <= 1.