A general three-series theorem
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- by B. M. Brown PDF
- Proc. Amer. Math. Soc. 28 (1971), 573-577 Request permission
Erratum: Proc. Amer. Math. Soc. 32 (1972), 634.
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
- J. L. Doob, Stochastic processes, John Wiley & Sons, Inc., New York; Chapman & Hall, Ltd., London, 1953. MR 0058896
- D. L. Burkholder, Martingale transforms, Ann. Math. Statist. 37 (1966), 1494–1504. MR 208647, DOI 10.1214/aoms/1177699141
Additional Information
- © Copyright 1971 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 28 (1971), 573-577
- MSC: Primary 60.30
- DOI: https://doi.org/10.1090/S0002-9939-1971-0277020-5
- MathSciNet review: 0277020