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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 

 

A general three-series theorem


Author: B. M. Brown
Journal: Proc. Amer. Math. Soc. 28 (1971), 573-577
MSC: Primary 60.30
DOI: https://doi.org/10.1090/S0002-9939-1971-0277020-5
Erratum: Proc. Amer. Math. Soc. 32 (1972), 634.
MathSciNet review: 0277020
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Abstract: Let $ \{ \Omega ,\mathcal{F},P\} $ be a probability space. The subset of $ \Omega $ on which an arbitrary sequence of random variables converges is shown to be equivalent to the intersection of three other sets, each specified by the almost sure convergence of a certain sequence of random variables. Kolmogorov's three-series theorem, which gives necessary and sufficient conditions for the almost sure convergence of a sequence of sums of independent random variables, is obtainable as a particular case of the present result.


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DOI: https://doi.org/10.1090/S0002-9939-1971-0277020-5
Keywords: Three-series theorem, independent random variables, conditional expectations, almost sure convergence, equivalent sets, martingale differences, martingale convergence theorem
Article copyright: © Copyright 1971 American Mathematical Society