Iterates of a product of conditional expectation operators
Let \((\Omega ,{\mathcal F},\mu )\) be a probability space and let \(T=P_1 P_2 \dots P_d\) be a finite product of conditional expectations with respect to the sub \(\sigma\)-algebras \({\mathcal F}_1 ,{\mathcal F}_2 ,\dots,{\mathcal F}_d \). In the paper it is shown that for all \(f\in L_p(\mu)\), \(1\leq p\leq 2\), the sequence \(\{ T^n f\}\) converges \(\mu\)-a.e., with \[ \lim_{n\to \infty}T^n f=E[f| {\mathcal F}_1\cap {\mathcal F}_2 \cap \dots\cap {\mathcal F}_d ]\quad \mu\text{-a.e.} \]
scientific article; zbMATH DE number 5116379