A universal form of the Chung-type law of the iterated logarithm
成果类型：
Article
署名作者：
Kesten, H
署名单位：
Cornell University
刊物名称：
ANNALS OF PROBABILITY
ISSN/ISSBN：
0091-1798
DOI：
10.1214/aop/1023481104
发表日期：
1997
页码：
1588-1620
关键词：
摘要：
Let {X-i}(i greater than or equal to 1) be i.i.d. random variables with common distribution function F-1 and let S-n = Sigma(1)(n) X-i. We find a necessary and sufficient condition (directly in terms of Fl for the existence of sequences of constants (a,) and {beta(n)} with beta(n) up arrow infinity such that 0 < lim inf beta(n)(-1) max(j less than or equal to n) /S-j-alpha(j)/ < infinity w.p.1., and such that for any choice of <(alpha)over tilde>(n), it hold w.p.1 that lim inf beta(n)(-1) max(j less than or equal to n) /S-j - <(alpha)over tilde>(j)/ > 0. The latter requirement is added to rule out sequences {beta(n)} which grow too fast and entirely overwhelm the fluctuations of S-n.