## The Annals of Probability

### A Law of Large Numbers for Identically Distributed Martingale Differences

John Elton

#### 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

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