Asymptotics of random sums of negatively dependent random variables in the presence of dominatedly varying tails

Abstract

Let X1, X2, . . . be a sequence of negatively dependent and identically distributed random variables, and let N be a counting random variable independent of Xi’s. In this paper, we study the asymptotics for the tail probability of the random sum \( {S_N} = \sum\nolimits_{k = 1}^N {{X_k}} \) in the presence of heavy tails. We consider the following three cases: (i) P(N > x) = o(P(X1 > x)), and the distribution function (d.f.) of X1 is dominatedly varying; (ii) P(X1 > x) = o(P(N > x)), and the d.f. of N is dominatedly varying; (iii) the tails of X1 and N are asymptotically comparable and dominatedly varying.