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.
Publié le : 1981-06-14
Classification:
Martingale,
law of large numbers,
maximal function,
almost sure convergence,
60F15,
60G45
@article{1176994414,
author = {Elton, John},
title = {A Law of Large Numbers for Identically Distributed Martingale Differences},
journal = {Ann. Probab.},
volume = {9},
number = {6},
year = {1981},
pages = { 405-412},
language = {en},
url = {http://dml.mathdoc.fr/item/1176994414}
}
Elton, John. A Law of Large Numbers for Identically Distributed Martingale Differences. Ann. Probab., Tome 9 (1981) no. 6, pp. 405-412. http://gdmltest.u-ga.fr/item/1176994414/