The Annals of Probability

A Law of Large Numbers for Identically Distributed Martingale Differences

John Elton

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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.

Article information

Source
Ann. Probab., Volume 9, Number 3 (1981), 405-412.

Dates
First available in Project Euclid: 19 April 2007

Permanent link to this document
https://projecteuclid.org/euclid.aop/1176994414

Digital Object Identifier
doi:10.1214/aop/1176994414

Mathematical Reviews number (MathSciNet)
MR614626

Zentralblatt MATH identifier
0463.60039

JSTOR
links.jstor.org

Subjects
Primary: 60F15: Strong theorems
Secondary: 60G45

Keywords
Martingale law of large numbers maximal function almost sure convergence

Citation

Elton, John. A Law of Large Numbers for Identically Distributed Martingale Differences. Ann. Probab. 9 (1981), no. 3, 405--412. doi:10.1214/aop/1176994414. https://projecteuclid.org/euclid.aop/1176994414


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