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June, 1981 A Law of Large Numbers for Identically Distributed Martingale Differences
John Elton
Ann. Probab. 9(3): 405-412 (June, 1981). DOI: 10.1214/aop/1176994414

Abstract

The averages of an identically distributed martingale difference sequence converge in mean to zero, but the almost sure convergence of the averages characterizes $L \log L$ in the following sense: if the terms of an identically distributed martingale difference sequence are in $L \log L$, the averages converge to zero almost surely; but if $f$ is any integrable random variable with zero expectation which is not in $L \log L$, there is a martingale difference sequence whose terms have the same distributions as $f$ and whose averages diverge almost surely. The maximal function of the averages of an identically distributed martingale difference sequence is integrable if its terms are in $L \log L$; the converse is false.

Citation

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John Elton. "A Law of Large Numbers for Identically Distributed Martingale Differences." Ann. Probab. 9 (3) 405 - 412, June, 1981. https://doi.org/10.1214/aop/1176994414

Information

Published: June, 1981
First available in Project Euclid: 19 April 2007

zbMATH: 0463.60039
MathSciNet: MR614626
Digital Object Identifier: 10.1214/aop/1176994414

Subjects:
Primary: 60F15
Secondary: 60G45

Keywords: Almost sure convergence , Law of Large Numbers , martingale , maximal function

Rights: Copyright © 1981 Institute of Mathematical Statistics

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Vol.9 • No. 3 • June, 1981
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