A Law of Large Numbers for Identically Distributed Martingale Differences

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.