Iterates of Conditional Expectation Operators

Part of the Selected Works in Probability and Statistics book series (SWPS)


Let {Tn} be a sequence of conditional expectation operators in \(L_1=L_1(W, \ F, \ P)\) where (W, F, P) is a probability space. Let \(S_n=T_n \ldots T_2 T_1\). It is known [l, p. 331] that if {T n } is monotone decreasing, that is, if the range of \(T_{n+1}\) is a subset of the range of \(T_n\) for all n, then for each x in L 1 the sequence {S n x} converges almost everywhere. Here, the pointwise convergence behavior of {Snx} is studied under other conditions. For example, if \(T_{2n-1}={T_1} \ {\rm and} \ {T_{2n}}={T_2}\) for all n, does {Snx} converge almost everywhere? This question was first posed by J. L. Doob. It is proved here that if x is in L2, then this is indeed the case, and, furthermore, \({\rm sup}_{n} \ |S_{n}x| \ {\rm is \ in} \ L_2\). Several of the preliminary results needed, especially Theorems 1 and 2, seem to be of some interest in their own right. The linear spaces mentioned in this paper may be either real or complex. All of our results hold with either interpretation.


Orthogonal Projection Strong Limit Monotone Convergence Theorem YORKTOWN Height Unbounded Component 


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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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