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Proceedings of the American Mathematical Society

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A general three-series theorem

Author: B. M. Brown
Journal: Proc. Amer. Math. Soc. 28 (1971), 573-577
MSC: Primary 60.30
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.

References [Enhancements On Off] (What's this?)

  • [1] J. L. Doob, Stochastic processes, Wiley, New York; Chapman & Hall, London, 1953. MR 15, 445. MR 0058896 (15:445b)
  • [2] D. L. Burkholder, Martingale transforms, Ann. Math. Statist. 37 (1966), 1494-1504. MR 34 #8456. MR 0208647 (34:8456)

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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

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