Tail behavior of negatively associated heavy-tailed sums

Abstract

Consider a sequence {Xk, k ≥ 1} of random variables on (−∞, ∞). Results on the asymptotic tail probabilities of the quantities , and S(n) = max0 ≤ k ≤ nSk, with X0 = 0 and n ≥ 1, are well known in the case where the random variables are independent with a heavy-tailed (subexponential) distribution. In this paper we investigate the validity of these results under more general assumptions. We consider extensions under the assumptions of having long-tailed distributions (the class L) and having the class D ∩ L, where D is the class of distribution functions with dominatedly varying tails. Some results are also given in the case where Xk, k ≥ 1, are not necessarily identically distributed and/or independent.