Tail behavior of negatively associated heavy-tailed sums

Abstract

Consider a sequence {X k, k ≥ 1} of random variables on (-∞, ∞). Results on the asymptotic tail probabilities of the quantities S n = ∑ k=0 n X k, X (n) = max 0≤k≤n X k, and S (n) = max 0≤k≤n S k, with X 0 = 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 X k, k ≥ 1, are not necessarily identically distributed and/or independent. © Applied Probability Trust 2006.