Characterization of the Cluster Set of the LIL Sequence in Banach Space

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.