Some Results on the Cluster Set $C\bigg(\bigg{\frac{S_n}{a_n}\bigg}\bigg)$ and the LIL

We investigate the cluster set $C(\{S_n/a_n\})$ under conditions necessary for the bounded law of the iterated logarithm, and obtain necessary and sufficient conditions for the LIL in spaces satisfying a certain comparison principle. In particular, these results settle some previously unanswered questions in the Hilbert space setting.