We investigate the relationship between the bounded law of the iterated logarithm and the compact law of the iterated logarithm for Banach space valued random variables through the cluster set of $S_n/a_n$. Some rate of escape questions are also examined.
"When is the Cluster Set of $S_n/a_n$ Empty?." Ann. Probab. 9 (3) 377 - 394, June, 1981. https://doi.org/10.1214/aop/1176994412