The Asymptotics For The Tail Probability Of Random Sums With Heavy-tailed Random Numbers

It is well known that the asymptotics for the tail probability of random sums have wide and important applications in teletraffic, risk theory, earthquake insurance, queuing theory, branching process theory and so on. Researchers have paid more attention to them and many results in this field have been obtained. Let{X, Xk, k≥1} be a sequence of random variables (r.v.'s) supported on R. Let T be a nonnegative integer-value r.v. which is independent of {X,Xk,k≥1}. Most of the time, people concentrated on the case of the tail probability of X is heavier than that of T. Recently, Fay et al (2006) began to focuss on the other case when the tail probability of T is heavier than that of X. They got the sufficient conditions and necessary conditions for random sums having regularly varying tail according a asymptotics equivalence of ST. Robert et al (2008), Aleskeviciene et al (2008) and Zhang et al (2009) extended their results from one aspect:under the condition of T is consistently varying, they got the asympotics equivalence P(ST>x)～P(T>x/EX), and then ST is consistently varying, this gives a sufficient condition for ST belonging to the class of consistent variation. Inspired by them, under certain conditions, for NOD r.v.'s{X,Xk,k≥1} supported on R, we get the same asympotics equivalence of ST, and then acquire the necessary and sufficient conditions for ST having consistently varying tail. We also obtain the asymptotic behavior of ST when T has long-tailed and dominated varying tail.