Uniform estimate for the tail probabilities of randomly weighted sums

Several authors have studied the uniform estimate for the tail probabilities of randomly weighted sums \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sum\limits_{i = 1}^n {\theta _i X_i }$$\end{document} and their maxima. In this paper, we generalize their work to the situation that {Xi, i ≥ 1} is a sequence of upper tail asymptotically independent random variables with common distribution from the class \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal{D} \cap \mathcal{L}$$\end{document}, and {θi, i ≥ 1} is a sequence of nonnegative random variables, independent of {Xi, i ≥ 1} and satisfying some regular conditions. Moreover, no additional assumption is required on the dependence structure of {θi, i ≥ 1}.