The Annals of Probability
- Ann. Probab.
- Volume 10, Number 3 (1982), 828-830.
Conditional Generalizations of Strong Laws Which Conclude the Partial Sums Converge Almost Surely
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.
Ann. Probab., Volume 10, Number 3 (1982), 828-830.
First available in Project Euclid: 19 April 2007
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Conditional Borel-Cantelli Lemma conditional three-series theorem conditional strong laws martingales arbitrarily-dependent sequences of random variables almost sure convergence of partial sums
Hill, T. P. Conditional Generalizations of Strong Laws Which Conclude the Partial Sums Converge Almost Surely. Ann. Probab. 10 (1982), no. 3, 828--830. doi:10.1214/aop/1176993792. https://projecteuclid.org/euclid.aop/1176993792