Precise asymptotics in the law of the iterated logarithm

• 成果类型: Article
• 署名作者: Gut, A; Spataru, A
• 署名单位: Uppsala University; Romanian Academy; Institute of Mathematical Statistics & Applied Mathematics of Romanian Academy
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• 发表日期: 2000
• 页码: 1870-1883
• 关键词: large numbers

摘要

Let X, X-1, X-2,... be i.i.d. random variables with mean 0 and positive, finite variance sigma (2), and set S-n = X-1 + ... + X-n, n greater than or equal to 1. Continuing earlier work related to strong laws, we prove the following analogs for the law of the iterated logarithm: lim(epsilon down arrow sigma root2) root epsilon (2)-2 sigma (2) Sigma (n greater than or equal to3) 1/n P(\S-n\ greater than or equal to epsilon rootn log log n + a(n)) = sigma root2 whenever a(n) = O(rootn(log log n)(-gamma)) for some gamma greater than or equal to 1/2 (assuming slightly more than finite variance), and lim(epsilon down arrow0)epsilon (2) Sigma (n greater than or equal to3) 1/n log n P(\S-n\ greater than or equal to epsilon rootn log log n) = sigma (2).