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

Let {X,X _n ; n ≥ 0} be a sequence of independent and identically distributed random variables with EX = 0, and assume that EX ² I(|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 \(\sum\nolimits_{n = 0}^\infty {{\beta ^n}{X_n}\left( {0 < \beta < 1} \right)} \) is obtained, under some minimal conditions.