Asymptotic Behavior For Random Sums Of Negatively Associated Random Variables

The random sums of random variables plays an important role in various applied probability fields such as insurance mathematics, queuing theory, teletraffic, etc, and nowadays it has already been studied and referred by an extensive literature. Let {X, X_k : k≥1} be a sequence of random variables, with the common distribution F(x) = 1 - (?)(x) = P(X≤x). In this paper we focus on the asymptotic behavior of random sums S_τ=∑_k=1^τX_k, whereτis an integer-valued nonnegative random variable independent of {X_k, k≥1}.In 2008, Ale(?)kevi(?)ien(?) et al presented a series of asymptotic relation describing the tail behavior of S_τin which the tail of X is not heavier than ofτ. However under the assumption that X andτis asymptotically equivalent they merely attained an asymptotic upper/lower bound of S_τwhich is far from accomplished. The present paper will step further in this kind of random sums' tail behavior, to give an equivalent version of asymptotic behavior of S_τin the case X andτis asymptotically equivalent, and extend the result to the negatively associated {X, X_k : k≥1}, finally fulfill the tail behavior of S_τwith an unified outcome.