Generalized LIL for geometrically weighted random series in Banach spaces

Let $\{X,X_n;n?0\}$ be a sequence of independent and identically distributed random variables, taking values in a separable Banach space B with topological dual $B^{?}$. Considering the geometrically weighted series $\xi (\beta )={\sum }_{n=0}^{\infty }{\beta }^n{X}_n$ for $0<\beta <1$, motivated by Einmahl and Li (2005, 2008), a general law of the iterated logarithm for $\xi (\beta )$ is established.