Abstract
Let $S_n = X_1 + \cdots + X_n$, where $X_1, X_2, \ldots$ are iid Banach-space-valued random variables with weak mean 0 and weak second moments. Let $K$ be the unit ball of the reproducing kernel Hilbert space associated to the covariance of $X$. We show that the cluster set of $\{S_n/(2n \log \log n)^{1/2}\}$ either is empty or has the form $\alpha K$, where $0 \leq \alpha \leq 1$. A series condition is given which determines the value of $\alpha$. In a companion paper, examples are given to show that all $\alpha \in \lbrack 0, 1 \rbrack$ do occur.
Citation
Kenneth S. Alexander. "Characterization of the Cluster Set of the LIL Sequence in Banach Space." Ann. Probab. 17 (2) 737 - 759, April, 1989. https://doi.org/10.1214/aop/1176991424
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