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 = n k=0 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.

Sums of Pairwise Quasi-Asymptotically Independent Random Variables with Consistent Variation

“…Substituting (12) and (13) into (11) and using the definition of the class , we obtain the desired upper asymptotic estimate…”

“…Geluk and Ng [11] , Tang [27,28] , and Ko and Tang [15] studied the asymptotic tail behavior of sums under some general dependence assumptions requiring that the random variables under consideration are negatively (or not too positively) dependent.…”

Tail Behavior of Sums and Maxima of Sums of Dependent Subexponential Random Variables

“…. , n, Geluk and Ng [9] proved relations (1.3) in the negative association structure. Tang [15] relaxed the dependence structure from negative association to pairwise negative quadrant dependence.…”

“…Recently, Geluk and De Vries [8] obtained similar results for independent but nonidentically distributed nonnegative r.v.s. Furthermore, in the independent structure, Geluk and Ng [9] proposed the extension of relation (1.2) to the case of the nonidentically distributed and not necessarily nonnegative r.v.s, complementing the estimates for the quantity S (n) : Let X k , k ≥ 1 be independent r.v.s with distributions F X k on (−∞, ∞), k = 1, 2, . .…”

“…Motivated by Ko and Tang [11] and Geluk and Ng [9], the purpose of this paper is simultaneously: (i) to extend the supports of the r. …”