Weak Laws of Large Numbers for sequences of random variables with infinite rth moments

Let (Xn;n≥1)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${(X_n;n \geq 1)}$$\end{document} be a sequence of independent random variables with infinite rth absolute moments for some 0<r<2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${0 < r < 2}$$\end{document}. We investigate weak laws of large numbers for the weighted sum Sn=∑j=1mncnjXj\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${S_n = \sum_{j=1}^{m_n}c_{nj}X_j}$$\end{document}, where (cnj;1≤j≤mn,n≥1)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${(c_{nj};1 \leq j \leq m_n,n \geq 1)}$$\end{document} is an array of real numbers. As illustrative examples, we obtain a weak law of large numbers of extended Pareto–Zipf distributions and generalized Feller Game. Furthermore, these results are applied to study weak law of large numbers of moving average sums of a sequence of i.i.d. random variables.