The Annals of Probability

Conditional Generalizations of Strong Laws Which Conclude the Partial Sums Converge Almost Surely

T. P. Hill

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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.

Article information

Source
Ann. Probab., Volume 10, Number 3 (1982), 828-830.

Dates
First available in Project Euclid: 19 April 2007

Permanent link to this document
https://projecteuclid.org/euclid.aop/1176993792

Digital Object Identifier
doi:10.1214/aop/1176993792

Mathematical Reviews number (MathSciNet)
MR659552

Zentralblatt MATH identifier
0486.60028

JSTOR
links.jstor.org

Subjects
Primary: 60F15: Strong theorems
Secondary: 60G45

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

Citation

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


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