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December, 1979 Markov-Dependent $\sigma$-Fields and Conditional Expectations
Richard Isaac
Ann. Probab. 7(6): 1088-1091 (December, 1979). DOI: 10.1214/aop/1176994905

Abstract

Basterfield showed that if $X \in L \log L$ and $\{\mathscr{F}_n\}$ form a sequence of independent $\sigma$-fields, then $E(X\mid\mathscr{F}_n)\rightarrow EX$ a.s. His proof uses the theory of Orlicz spaces. We generalize Basterfield's theorem to the case of Markov-dependent $\sigma$-fields and also weaken the restrictions on $X$. Our approach is different from Basterfield's in that it is martingale-theoretic.

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Richard Isaac. "Markov-Dependent $\sigma$-Fields and Conditional Expectations." Ann. Probab. 7 (6) 1088 - 1091, December, 1979. https://doi.org/10.1214/aop/1176994905

Information

Published: December, 1979
First available in Project Euclid: 19 April 2007

zbMATH: 0421.60029
MathSciNet: MR548906
Digital Object Identifier: 10.1214/aop/1176994905

Subjects:
Primary: 60F15
Secondary: 60G45 , 60J05

Keywords: $\sigma$-field , Markov-dependent , martingale

Rights: Copyright © 1979 Institute of Mathematical Statistics

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Vol.7 • No. 6 • December, 1979
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