On the Almost Sure Behavior of Sums of IID Random Variables in Hilbert Space

Abstract

We study the almost sure behavior of sums of $\operatorname{iid}$ random variables satisfying the bounded LIL in Hilbert space. We show that the almost sure behavior is different from the Gaussian case, whenever the second strong moments are infinite. A law of the $k$ times iterated logarithm is established which refines the bounded LIL. The interesting feature here is that contrary to the known conditions for the bounded LIL, one needs not only moment type conditions but also a nice structure of the covariance operator.