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April, 1989 Characterization of the Cluster Set of the LIL Sequence in Banach Space
Kenneth S. Alexander
Ann. Probab. 17(2): 737-759 (April, 1989). DOI: 10.1214/aop/1176991424


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.


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Kenneth S. Alexander. "Characterization of the Cluster Set of the LIL Sequence in Banach Space." Ann. Probab. 17 (2) 737 - 759, April, 1989.


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

zbMATH: 0679.60006
MathSciNet: MR985387
Digital Object Identifier: 10.1214/aop/1176991424

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. 2 • April, 1989
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