STRONG RENEWAL THEOREMS WITH INFINITE MEAN BEYOND LOCAL LARGE DEVIATIONS

• 成果类型: Article
• 署名作者: Chi, Zhiyi
• 署名单位: University of Connecticut
• 刊物名称: ANNALS OF APPLIED PROBABILITY
• ISSN/ISSBN: 1050-5164
• DOI: 10.1214/14-AAP1029
• 发表日期: 2015
• 页码: 1513-1539
• 关键词: random-walks

摘要

Let F be a distribution function on the line in the domain of attraction of a stable law with exponent alpha is an element of (0,1/2]. We establish the strong renewal theorem for a random walk S-1, S-2, ... with step distribution F, by extending the large deviations approach in Doney [Probab. Theory Related Fileds 107 (1997) 451-465]. This is done by introducing conditions on F that in general rule out local large deviations bounds of the type P{S-n is an element of(x, x + h]} = O(n)(F) over bar (x)/x, hence are significantly weaker than the boundedness condition in Doney (1997). We also give applications of the results on ladder height processes and infinitely divisible distributions.