Suppose that for every independent sequence of random variables satisfying some hypothesis condition $H$, it follows that the partial sums converge almost surely. Then it is shown that for every arbitrarily-dependent sequence of random variables, the partial sums converge almost surely on the event where the conditional distributions (given the past) satisfy precisely the same condition $H$. Thus many strong laws for independent sequences may be immediately generalized into conditional results for arbitrarily-dependent sequences.
Publié le : 1982-08-14
Classification:
Conditional Borel-Cantelli Lemma,
conditional three-series theorem,
conditional strong laws,
martingales,
arbitrarily-dependent sequences of random variables,
almost sure convergence of partial sums,
60F15,
60G45
@article{1176993792,
author = {Hill, T. P.},
title = {Conditional Generalizations of Strong Laws Which Conclude the Partial Sums Converge Almost Surely},
journal = {Ann. Probab.},
volume = {10},
number = {4},
year = {1982},
pages = { 828-830},
language = {en},
url = {http://dml.mathdoc.fr/item/1176993792}
}
Hill, T. P. Conditional Generalizations of Strong Laws Which Conclude the Partial Sums Converge Almost Surely. Ann. Probab., Tome 10 (1982) no. 4, pp. 828-830. http://gdmltest.u-ga.fr/item/1176993792/