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

Let X1,X2, ... be a sequence of dependent and heavy-tailed random variables with distributions F1, F2, ... on (−∞,∞), and let τ be a nonnegative integer-valued random variable independent of the sequence {Xk, k ≥ 1}. In this framework, the asymptotic behavior of the tail probabilities of the quantities \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$S_n = \sum\limits_{k = 1}^n {X_k }$$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$S_{(n)} = \mathop {\max }\limits_{1 \leqslant k \leqslant n} S_k$$\end{document} for n > 1, and their randomized versions Sτ and S(τ) are studied. Some applications to the risk theory are presented.