A self-normalized law of the iterated logarithm for the geometrically weighted random series

Let {X,Xn; n ≥ 0} be a sequence of independent and identically distributed random variables with EX = 0, and assume that EX2I(|X| ≤ x) is slowly varying as x→∞, i.e., X is in the domain of attraction of the normal law. In this paper, a self-normalized law of the iterated logarithm for the geometrically weighted random series \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sum\nolimits_{n = 0}^\infty {{\beta ^n}{X_n}\left( {0 < \beta < 1} \right)} $$\end{document} is obtained, under some minimal conditions.