  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.
