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Successive Conditional Expectations of an Integrable Function

  • Burgess Davis
  • Renming Song
Chapter
Part of the Selected Works in Probability and Statistics book series (SWPS)

Abstract

Rota [5] has shown recently that if {T n } is a sequence of conditional expectation operators
$${S_n} = {T_0}{T_1} \ldots {T_{n - 1}}{T_n}{T_{n - 1}} \ldots {T_1}{T_0},$$
(1)
and X is a random variable such that
$$E\left( {|X|{{\log }^ + }|X|} \right) < \infty, $$
(2)
then the sequence {S n X} converges almost everywhere to an integrable function.

Keywords

Independent Random Variable Conditional Expectation Monotone Convergence Theorem Continuous Distribution Function Real Number Sequence 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

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    Blackwell, David and Dubins, Lester E. A converse to the dominated conver gence theorem. Illinois J. Math.To appearGoogle Scholar
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    BurkholderD. L. and ChowY. S. (1961). Iterates of conditional expectation opera tors. Proc. Amer. Math. Soc.12 490-495.MATHCrossRefMathSciNetGoogle Scholar
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    DoobJ. L. (1953). Stochastic Processes. Wiley, New York.MATHGoogle Scholar
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    HardyG. H. and LittlewoodJ. E. (1930). A maximal theorem with function-theoretic applications. Acta Math.54 81-116.CrossRefMathSciNetGoogle Scholar
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    Rota, Gian-Carlo (1962). An "alternierende Verfahren" for general positive operators. Bull. Amer. Math. Soc.68 95-102.MATHCrossRefMathSciNetGoogle Scholar
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    SteinE. M. (1961). On the maximal ergodic theorem. Proc. Nat. Acad. Sei. USA.47 1894-1897.MATHCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2011

Authors and Affiliations

  1. 1.Department of Mathematics and Department of StatisticsPurdue UniversityWest LafayetteUSA
  2. 2.Department of MathematicsUniversity of IllinoisUrbanaUSA

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