Asymptotic results for tail probabilities of sums of dependent heavy-tailed random variables

TL;DR

This paper investigates the asymptotic tail probabilities of sums and maxima of dependent heavy-tailed random variables, including randomized versions, with applications to risk processes.

Contribution

It provides new asymptotic results for tail probabilities of dependent heavy-tailed variables and their maxima, extending existing theory to randomized sums and maxima.

Findings

Asymptotic behavior of tail probabilities for maxima and sums of dependent heavy-tailed variables.

Results for randomized versions involving a nonnegative integer-valued random variable.

Applications to risk processes demonstrate practical relevance.

Abstract

Let $\{X_1, X_2, ... \}$ be a sequence of dependent heavy-tailed random variables with distributions $F_1, F_2,...$ on $(-\infty,\infty)$, and let $\tau$ be a nonnegative integer-valued random variable independent of the sequence $\{X_k, k \ge 1\}$. In this framework, we study the asymptotic behavior of the tail probabilities of the quantities $X_{(n)} = \max_{1\le k \le n} X_k$, $S_n =\sum_{k=1}^n X_k$ and $S_{(n)}=\max_{1\le k\le n} S_k$ for $n>1$, and for those of their randomized versions $X_{(\tau)}$, $S_{\tau}$ and $S_{(\tau)}$. We also consider applications of the results obtained to some commonly-used risk processes.