Approximation of the Tail Probability of Dependent Random Sums Under Consistent Variation and Applications

In this paper, we consider the random sums of one type of asymptotically quadrant sub-independent and identically distributed random variables {X, X _i , i?=?1, 2,???} with consistently varying tails. We obtain the asymptotic behavior of the tail $\textsf{P}(X_1+\cdots+X_\eta>x)$ under different cases of the interrelationships between the tails of X and η, where η is an integer-valued random variable independent of {X, X _i , i?=?1, 2,???}. We find out that the asymptotic behavior of $\textsf{P}(X_1+\cdots+X_\eta>x)$ is insensitive to the dependence assumed in the present paper. We state some applications of the asymptotic results to ruin probabilities in the compound renewal risk model under dependent risks. We also state some applications to a compound collective risk model under the Markov environment.