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Proceedings of the American Mathematical Society

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An extension of Skorohod's almost sure representation theorem


Authors: David Blackwell and Lester E. Dubins
Journal: Proc. Amer. Math. Soc. 89 (1983), 691-692
MSC: Primary 60B10
MathSciNet review: 718998
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Abstract: Skorohod discovered that if a sequence $ {Q_n}$ of countably additive probabilities on a Polish space converges in the weak star topology, then, on a standard probability space, there are $ {Q_n}$-distributed $ {f_n}$ which converge almost surely. This note strengthens Skorohod's result by associating, with each probability $ Q$ on a Polish space, a random variable $ {f_Q}$ on a fixed standard probability space so that for each $ Q$, (a) $ {f_Q}$ has distribution $ Q$ and (b) with probability 1, $ {f_P}$ is continuous at $ P = Q$.


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

  • [1956] A. V. Skorohod, Limit theorems for stochastic processes, Teor. Veroyatnost. i Primenen. 1 (1956), 289–319 (Russian, with English summary). MR 0084897

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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9939-1983-0718998-0
Keywords: Probability, almost sure convergence, weak convergence
Article copyright: © Copyright 1983 American Mathematical Society