Asymptotics of randomly stopped sums in the presence of heavy tails

被引:38
|
作者
Denisov, Denis [1 ,2 ]
Foss, Serguei [2 ,3 ]
Korshunov, Dmitry [3 ]
机构
[1] Cardiff Univ, Sch Math, Cardiff CF24 4AG, S Glam, Wales
[2] Heriot Watt Univ, Sch MACS, Edinburgh EH14 4AS, Midlothian, Scotland
[3] Sobolev Inst Math, Novosibirsk 630090, Russia
基金
英国工程与自然科学研究理事会;
关键词
convolution equivalence; heavy-tailed distribution; random sums of random variables; subexponential distribution; upper bound; RANDOM TIME-INTERVAL; LARGE DEVIATIONS; LOWER LIMITS; RANDOM-WALK; DISTRIBUTIONS; THEOREMS; SUBEXPONENTIALITY; EQUIVALENCES; BEHAVIOR;
D O I
10.3150/10-BEJ251
中图分类号
O21 [概率论与数理统计]; C8 [统计学];
学科分类号
020208 ; 070103 ; 0714 ;
摘要
We study conditions under which P{S-tau > x} similar to P{M-tau > x} similar to E tau P{xi(1) > x} as x -> infinity, where S-tau is a sum xi(1) + ... + xi(tau) of random size tau and M-tau is a maximum of partial sums M-tau = max(n <=tau) S-n. Here, xi(n), n= 1, 2, ... , are independent identically distributed random variables whose common distribution is assumed to be subexponential. We mostly consider the case where tau is independent of the summands; also, in a particular situation, we deal with a stopping time. We also consider the case where E xi > 0 and where the tail of tau is comparable with, or heavier than, that of xi, and obtain the asymptotics P{S-tau > x) similar to E tau P{xi(1) > x} + P{tau > x/E xi} as x -> infinity. This case is of primary interest in branching processes. In addition, we obtain new uniform (in all x and n) upper bounds for the ratio P{S-n > x}/P{xi(1) > x} which substantially improve Kesten's bound in the subclass S* of subexponential distributions.
引用
收藏
页码:971 / 994
页数:24
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