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

Abstract

This article investigates the tail asymptotic behavior of the sum of pairwise quasi-asymptotically independent random variables with consistently varying tails. We prove that the tail probability of the sum is asymptotically equal to the sum of individual tail probabilities. This matches a feature of subexponential distributions. This result is then extended to weighted sums and random sums.

Uniform tail asymptotics for the stochastic present value of aggregate claims in the renewal risk model

“…Note that the result of Resnick and Willekens (1991) has recently been extended in many ways by Goovaerts Chen and Yuen (2009). Therefore, starting with these extended results, it should be possible and routine, but rather laborious, to further extend Theorem 2.1 to a somewhat broader class of heavy-tailed distributions (for example, the class of distributions with extended regularly varying tails), and to the case that claim sizes possess a certain dependence structure (for example, pairwise asymptotic independence).…”

Uniform Asymptotics for the Finite-Time Ruin Probability in a General Risk Model With Pairwise Quasi-Asymptotically Independent Claims and Constant Interest Force

“…It should be mentioned that the idea of this proof is mainly from that of Chen and Yuen ( [2]). Relation (3.3) is trivial if n = 1.…”

“…where the method of the division comes from Chen and Yuen ( [2]). For K 21 , by F ∈ C ⊂ D and Lemma 3.1(1), we have…”

“…Hence, in this paper we will consider a risk model with the claim sizes following a certain dependence structure, which was introduced by Chen and Yuen ( [2]) and Yi et al ([17]). Definition 1.1.…”

The Ultimate Ruin Probability of a Dependent Delayed-Claim Risk Model Perturbed by Diffusion With Constant Force of Interest

“…Also, Chen and Yuen [2] proposed a more general dependence structure below. Say that r.v.s {ξ i , i ≥ 1} are pairwise quasi-asymptotically independent (PQAI), if for any 1 ≤ i = j < ∞,…”