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July, 1989 Unusual Cluster Sets for the LIL Sequence in Banach Space
Kenneth S. Alexander
Ann. Probab. 17(3): 1170-1185 (July, 1989). DOI: 10.1214/aop/1176991263


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.


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Kenneth S. Alexander. "Unusual Cluster Sets for the LIL Sequence in Banach Space." Ann. Probab. 17 (3) 1170 - 1185, July, 1989.


Published: July, 1989
First available in Project Euclid: 19 April 2007

zbMATH: 0694.60005
MathSciNet: MR1009451
Digital Object Identifier: 10.1214/aop/1176991263

Primary: 60B12
Secondary: 60F15

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

Rights: Copyright © 1989 Institute of Mathematical Statistics

Vol.17 • No. 3 • July, 1989
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