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February, 1979 A Strong Law of Large Numbers for Subsequences of Random Elements in Separable Banach Spaces
A. Bozorgnia, M. Bhaskara Rao
Ann. Probab. 7(1): 156-158 (February, 1979). DOI: 10.1214/aop/1176995157

Abstract

In 1967, Komlos proved that if $\{\xi_n\}$ is a sequence of real random variables for which $\sup_{n \geqslant 1}E|\xi_n| < \infty$, then there exists a subsequence $\{\eta_n\}$ of $\{\xi_n\}$ and an integrable random variable $\eta$ such that for an arbitrary subsequence $\{\eta_n\}$ of $\{\eta_n\}$. $$\lim_{n \rightarrow \infty} \frac{1}{n}(\check{\eta}_1 + \check{\eta}_2 + \cdots + \check{\eta}_n) = \eta \mathrm{a.s.}$$ In this paper, we attempt to extend this result to separable Banach space valued random elements. We impose a condition stronger than uniform integrability.

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A. Bozorgnia. M. Bhaskara Rao. "A Strong Law of Large Numbers for Subsequences of Random Elements in Separable Banach Spaces." Ann. Probab. 7 (1) 156 - 158, February, 1979. https://doi.org/10.1214/aop/1176995157

Information

Published: February, 1979
First available in Project Euclid: 19 April 2007

zbMATH: 0392.60011
MathSciNet: MR515822
Digital Object Identifier: 10.1214/aop/1176995157

Subjects:
Primary: 60F15
Secondary: 60B05

Keywords: Random elements in separable Banach spaces , Strong law of large numbers

Rights: Copyright © 1979 Institute of Mathematical Statistics

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Vol.7 • No. 1 • February, 1979
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