Precise asymptotics in Spitzer and Baum-Katz's law of large numbers: the semistable case

[摘要] Let X-1, X-2,... be i.i.d. random variables with distribution mu and with mean zero, whenever the mean exists. Set S-n = X-1 + - - - + X-n. In recent years precise asymptotics as epsilon down arrow 0 have been proved for sums like Sigma(n=1)(infinity) n(-1)P {\S-n\ greater than or equal to epsilonn(1/p)}, assuming that mu belongs to the (normal) domain of attraction of a stable law. Our main results generalize these results to distributions mu belonging to the (normal) domain of semistable attraction of a semistable law. Furthermore, a limiting case new even in the stable situation is presented. (C) 2003 Elsevier Inc. All rights reserved.

[发布日期] 2003-12-01

[关键词] Tail probabilities of sums of i.i.d. random variables;semistable distributions;Spitzer's law;Baum-Katz's law