Tail probability of a random sum with a heavy-tailed random number and dependent summands

摘要

Let {xi, xi(k) : k >= 1} be a sequence of widely orthant dependent random variables with common distribution F satisfying E xi > 0. Let tau be a nonnegative integer-valued random variable. In this paper, we discuss the tail probabilities of random sums S tau = Sigma(tau)(n=1) when the random number tau has a heavier tail than the summands, i.e. P(xi > x)/P(tau > x) -> 0 as x -> infinity. Under some additional technical conditions, we prove that if tau has a consistently varying tail, then S-tau has a consistently varying tail and P(S-tau > x) similar to P(tau > x/E xi). On the other hand, the converse problem is also equally interesting. We prove that if S-tau has a consistently varying tail, then T has a consistently varying tail and that P(S-tau > x) similar to P(tau > x/E xi) still holds. In particular, the random number tau is not necessarily assumed to be independent of the summands {xi(k), : k >= 1} in Theorem 3.1 and Theorem 3.3. Finally, some applications to the asymptotic behavior of the finite-time ruin probabilities in some insurance risk models are given. (C) 2015 Elsevier Inc. All rights reserved.