Abstract. Let $\{X,X_n;n\ge0\} $ be a sequence of $B$-valued independent random variables. We consider the random variables $\xi(\beta )=\sum_{n=0}^\infty\beta ^nX_n$ for $0<\beta < 1$, and establish the LIL for $\xi(\beta )$ as $\beta\nearrow 1$. In particular, in the case when $\{X,X_n;n\ge0\} $ are independent and identically distributed, we prove that the LIL for $\xi(\beta )$ holds if and only if the same result is true for $S_n$, where $S_n=X_1+\cdots +X_n$, $n\ge1$. We also prove a central limit theorem for $\xi(\beta )$.

AMS Subject Classification (1991): 60F05, 60F15

Received June 12, 1996 and in revised form May 15, 1997. (Registered under 5791/2009.)