Some Properties For Distributions Of Random Sums

It is well known that the random sums of random variables have wide and important applications in queueing theory,risk theory, teletraffic,infinite divisibility theory,branching process theory and so on.Recently researchers have paid more attentions to them and many results in this field have been obtained.Let{X, Xk:k≥1} be a sequence of random variables supported on R or R+,with the common d.f. F(x)=P(X≤x).Denote by G=F*T the distribution of the random sum ST=X1+…+XT,where T is a nonnegative integer-valued random variable with the d.f. In this paper we study the prop-erties for distributions of random sums on two dimensions. On the one hand,we study the distributions of random sums with heavy-tailed terms and obtain a sufficient condition for G*2(x)～(?)(x) when the tail of X is lighter than that of T,that is,F(x)=o(H(x)).On the other hand, based on Denisov et al.(2008),Denisov et al.(2008) and Yu et al.(2008), we study the lower limits of the ratios of tails and some elementary results have been obtained,where F is a distribution on R.Further,we give some local versions and density versions of above results.