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August, 1982 Conditional Generalizations of Strong Laws Which Conclude the Partial Sums Converge Almost Surely
T. P. Hill
Ann. Probab. 10(3): 828-830 (August, 1982). DOI: 10.1214/aop/1176993792

Abstract

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.

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T. P. Hill. "Conditional Generalizations of Strong Laws Which Conclude the Partial Sums Converge Almost Surely." Ann. Probab. 10 (3) 828 - 830, August, 1982. https://doi.org/10.1214/aop/1176993792

Information

Published: August, 1982
First available in Project Euclid: 19 April 2007

zbMATH: 0486.60028
MathSciNet: MR659552
Digital Object Identifier: 10.1214/aop/1176993792

Subjects:
Primary: 60F15
Secondary: 60G45

Keywords: almost sure convergence of partial sums , arbitrarily-dependent sequences of random variables , Conditional Borel-Cantelli Lemma , conditional strong laws , conditional three-series theorem , Martingales

Rights: Copyright © 1982 Institute of Mathematical Statistics

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Vol.10 • No. 3 • August, 1982
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