The Annals of Probability

Unusual Cluster Sets for the LIL Sequence in Banach Space

Kenneth S. Alexander

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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.

Article information

Ann. Probab., Volume 17, Number 3 (1989), 1170-1185.

First available in Project Euclid: 19 April 2007

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier


Primary: 60B12: Limit theorems for vector-valued random variables (infinite- dimensional case)
Secondary: 60F15: Strong theorems

Law of the iterated logarithm cluster set Banach-space-valued random variables


Alexander, Kenneth S. Unusual Cluster Sets for the LIL Sequence in Banach Space. Ann. Probab. 17 (1989), no. 3, 1170--1185. doi:10.1214/aop/1176991263.

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