Asymptotics for the Tail Probability of Random Sums with a Heavy-Tailed Random Number and Extended Negatively Dependent Summands

Let {X,X _k : k ≥ 1} be a sequence of extended negatively dependent random variables with a common distribution F satisfying EX > 0. Let τ be a nonnegative integer-valued random variable, independent of {X,X _k : k ≥ 1}. In this paper, the authors obtain the necessary and sufficient conditions for the random sums $S_\tau = \sum\limits_{n = 1}^\tau {X_n } $ to have a consistently varying tail when the random number τ has a heavier tail than the summands, i.e., $\frac{{P(X > x)}} {{P(\tau > x)}} \to 0 $ as x→∞.