Some results on two-sided LIL behavior

• 成果类型: Article
• 署名作者: Einmahl, U; Deli, L
• 署名单位: Vrije Universiteit Brussel; Lakehead University
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• DOI: 10.1214/009117905000000198
• 发表日期: 2005
• 页码: 1601-1624
• 关键词: iterated logarithm; universal law; sums; attraction; variables; domain

摘要

Let {X, X-n; n > 1} be a sequence of i.i.d. mean-zero random variables, and let S-n = Sigma(n)(i) (=1) X-i, n >= 1. We establish necessary and sufficient conditions for having with probability 1, 0 < lim sup(n ->infinity)vertical bar Sn vertical bar/root nh(n) < infinity where h is from a suitable subclass of the positive, nondecreasing slowly varying functions. Specializing our result to h(n) = (log log n)(P), where p > 1 and to h(n) = (log n)(r), r > 0, we obtain analogues of the Hartman-Wintner LIL in the infinite variance case. Our proof is based on a general result dealing with LIL behavior of the normalized sums {S-n/C-n ; n >= 1}, where c, is a sufficiently regular normalizing sequence.