Successive Conditional Expectations of an Integrable Function

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


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


Independent Random Variable Conditional Expectation Monotone Convergence Theorem Continuous Distribution Function Real Number Sequence 
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© 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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