Markov-Dependent $\sigma$-Fields and Conditional Expectations
Isaac, Richard
Ann. Probab., Tome 7 (1979) no. 6, p. 1088-1091 / Harvested from Project Euclid
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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.
Publié le : 1979-12-14
Classification:  Markov-dependent,  $\sigma$-field,  martingale,  60F15,  60G45,  60J05 
@article{1176994905,
     author = {Isaac, Richard},
     title = {Markov-Dependent $\sigma$-Fields and Conditional Expectations},
     journal = {Ann. Probab.},
     volume = {7},
     number = {6},
     year = {1979},
     pages = { 1088-1091},
     language = {en},
     url = {http://dml.mathdoc.fr/item/1176994905}
}

Isaac, Richard. Markov-Dependent $\sigma$-Fields and Conditional Expectations. Ann. Probab., Tome 7 (1979) no. 6, pp.  1088-1091. http://gdmltest.u-ga.fr/item/1176994905/