A general LIL for B-valued geometrically weighted series under dependent assumption

Let {X,Xn; n≧0} be a sequence of identically distributed ψ-mixing dependent random variables taking values in a type 2 Banach space B with topological dual B∗. Considering the geometrically weighted series \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\xi(\beta)= \sum_{n=0}^{\infty}\beta^{n}X_{n}$\end{document} for 0<β<1, a general law of the iterated logarithm for ξ(β) is obtained without second moment.