Tail asymptotics of randomly weighted sums of dependent strong subexponential random variables

Following [C. Yu, Y. Wang, and D. Cheng, Tail behavior of the sums of dependent and heavy-tailed random variables, J. Korean Stat. Soc., 44(1):12–27, 2015], we revisit the asymptotic formulas of tail probabilities of randomly weighted sums and their maxima. Concretely, let {Xn, n ≥ 1} be a sequence of nonnegative random variables with common strong subexponential distribution, which are dependent according to a pretty weak structure, and let {θn, n ≥ 1} be another sequence of nonnegative and arbitrarily dependent random variables but independent of {Xn, n ≥ 1}. For any fixed n ≥ 1, under some mild conditions, we derive the asymptotics of the tail probability of randomly weighted sums Snθ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ {S}_n^{\theta } $$\end{document} = ∑i=1nθiXi\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ {\sum}_{i=1}^n{\theta}_i{X}_i $$\end{document} and their maxima Mnθ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ {M}_n^{\theta } $$\end{document} = max1≤m≥nSmθ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ {S}_m^{\theta } $$\end{document}. Our result substantially extends some existing ones in the literature.