A nonclassical LIL for geometrically weighted series in Banach space

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\nolimits_{n = 0}^\infty {\beta ^n X_n } $ for 0 < β < 1, and a sequence of positive constants {h(n), n ≥ 1}, which is monotonically approaching infinity and not asymptotically equivalent to log log n, a limit result for $\frac{{\sqrt {1 - \beta ^2 } \left\| {\xi (\beta )} \right\|}} {{\sqrt {2h(1/(1 - \beta ^2 ))} }} $ is achieved.