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

Let {ξ,ξk:k≥1} be a sequence of widely orthant dependent random variables with common distribution F satisfying Eξ>0. Let τ be a nonnegative integer-valued random variable. In this paper, we discuss the tail probabilities of random sums Sτ=∑n=1τξn when the random number τ has a heavier tail than the summands, i.e. P(ξ>x)/P(τ>x)→0 as x→∞. Under some additional technical conditions, we prove that if τ has a consistently varying tail, then Sτ has a consistently varying tail and P(Sτ>x)∼P(τ>x/Eξ). On the other hand, the converse problem is also equally interesting. We prove that if Sτ has a consistently varying tail, then τ has a consistently varying tail and that P(Sτ>x)∼P(τ>x/Eξ) still holds. In particular, the random number τ is not necessarily assumed to be independent of the summands {ξk: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.