Randomly weighted sums of subexponential random variables with application to capital allocation

We are interested in the tail behavior of the randomly weighted sum ∑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}$\end{document}, in which the primary random variables X1, …, Xn are real valued, independent and subexponentially distributed, while the random weights 𝜃1, …, 𝜃n are nonnegative and arbitrarily dependent, but independent of X1, …, Xn. For various important cases, we prove that the tail probability of ∑i=1n𝜃iXi\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\sum _{i=1}^{n}\theta _{i}X_{i}$\end{document} is asymptotically equivalent to the sum of the tail probabilities of 𝜃1X1, …, 𝜃nXn, which complies with the principle of a single big jump. An application to capital allocation is proposed.