Asymptotic Behavior of Weighted Sums of Weakly Negative Dependent Random Variables

Let be a sequence of weakly negative dependent (denoted by, WND) random variables with common distribution function F and let be other sequence of positive random variables independent of and for some and for all . In this paper, we study the asymptotic behavior of the tail probabilities of the maximum, weighted sums, randomly weighted sums and randomly indexed weighted sums of heavy-tailed weakly negative dependent random variables, say, , , , and , respectively, where are bounded positive real numbers and N is a nonnegative integer-valued random variables, independent of and for all . In fact, for a large class of heavy-tailed distribution functions, we show that the asymptotic relations,
,
hold as Finally, if and also is a sequence of identical independent positive random variables, then we prove that