A Law of the Iterated Logarithm for Random Geometric Series

Abstract

We consider the random variables $\xi(\beta) = \sum^\infty_{n = 0}\beta^n\varepsilon_n$ for $\beta < 1$. We prove that if the $\varepsilon_n$ are i.i.d. random variables with mean zero and variance 1, then a law of the iterated logarithm holds in the sense that the cluster set of $\frac{\sqrt{1 - \beta^2}}{2\log\log(1/(1 - \beta^2))}\xi(\beta),$ when $\beta$ converges to one, is the interval $\lbrack-1, 1\rbrack$.