Unusual Cluster Sets for the LIL Sequence in Banach Space

Abstract

Let $S_n = X_1 + \cdots + X_n$, where $X_1, X_2, \cdots$ 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 $X$. The cluster set $A$ of $\{S_n/(2n \log \log n)^{1/2}\}$ is known to be a.s. either empty or have form $\alpha K$, with $0 \leq \alpha \leq 1$ determined by a series condition. To show that this series condition is a complete characterization of $A$, examples are given to show that all $\alpha \in \lbrack 0, 1)$ do occur; $A = \phi$ and $\alpha = 1$ are already known possibilities. A regularity condition is given under which $A$ must be either $\phi$ or $K$. Under stronger moment conditions, a natural necessary and sufficient condition for $A = \phi$ is given.