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

Abstract

In this paper, we consider dependent random variables X k , k = 1, 2, . . . with supports on [−b k , ∞), respectively, where the b k ≥ 0 are some finite constants. We derive asymptotic results on the tail probabilities of the quantities S n = n k=1 X k , X (n) = max 1≤k≤n X k and S (n) = max 1≤k≤n S k , n ≥ 1 in the case where the random variables are dependent with heavy-tailed (subexponential) distributions, which substantially generalize the results of Ko and Tang (J. Appl. Probab. 45, 85-94, 2008).

A note on the tail behavior of randomly weighted and stopped dependent sums

“…, similar results were obtained in [5,14,20,24]. For the asymptotic results for tail probabilities of (weighted) sums of dependent subexponential r.v.s, see [22,25]. We now introduce the following assumption.…”

On the max-sum equivalence in presence of negative dependence and heavy tails

“…Motivated by the paper of Li and Tang [1] (see also [2]), the aim of this note is to investigate the equivalence among the quantities P(𝑆 𝑛 > 𝑥) , P(max{𝑋 1 , … , 𝑋 𝑛 } > 𝑥) , P(𝑆 (𝑛) > 𝑥) and ∑ 𝑛 𝑘=1 P(𝑋 𝑘 > 𝑥) under some dependence assumption on 𝑋 1 , … , 𝑋 𝑛 with nonidentical distributions. Comparing with previous results (see, e.g., [3], [4], [5], [6], [7], [8], [9]), we aim to restrict some conditions to the (heavy-tailed) distribution of 𝑋 (𝑛) : = max(𝑋 1 , … , 𝑋 𝑛 ) . The assumption that the r.v.s 𝑋 1 , … , 𝑋 𝑛 are nonidentically distributed is important for insurance mathematics, because the result can be applied to some risk models with insurance and financial risks.…”

“…(8) (ii) If 𝐺 𝑛 ∈ 𝒞 and 𝐹 𝑖 (−𝑥) = 𝑜(𝐹 𝑖 (𝑥)) for 𝑖 = 1, … , 𝑛, then P(𝑆 (𝑛) > 𝑥) ≥ P(𝑆 𝑛 > 𝑥) ≳ 𝐺 𝑛 (𝑥). (9)(iii) If 𝐺 𝑛 ∈ ℒ ∩ 𝒟 and 𝐹 𝑖 (𝐴) = 0 for some finite 𝐴 < 0, 𝑖 = 1, … , 𝑛, then relations in(9) hold. Proof.…”