Markov-Dependent $\sigma$-Fields and Conditional Expectations

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.