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

Let $\{X_1,X_2,\mathrm{.}\mathrm{.}\mathrm{.}\}$ be a sequence of dependent heavy-tailed random variables with distributions $F_1,F_2,\mathrm{.}\mathrm{.}\mathrm{.}$ on $(-\mathrm{\infty },\mathrm{\infty })$, 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$n>1$,andforthoseoftheirrandomizedversions$X_{(\tau)}$,$S_{\tau}$and$S_{(\tau)}$. We also consider applications of the results obtained to some commonly-used risk processes.