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 .(.)=..n=0.nXn for 0 < . < 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 1..2...(.).2h(1/(1..2)). is achieved.