Iterates of Conditional Expectations: Convergence and Divergence

Guolie Lan, Ze-Chun Hu, Wei Sun

Published 2019-03-10Version 1

In this paper, we investigate the convergence and divergence of iterates of conditional expectation operators. We show that if $(\Omega,\cal{F},P)$ is a non-atomic probability space, then divergent iterates of conditional expectations involving 3 or 4 sub-$\sigma$-fields of $\cal{F}$ can be constructed for a large class of random variables in $L^2(\Omega,\cal{F},P)$. This settles in the negative a long-open conjecture. On the other hand, we show that if $(\Omega,\cal{F},P)$ is a purely atomic probability space, then iterates of conditional expectations involving any finite set of sub-$\sigma$-fields of $\cal{F}$ must converge for all random variables in $L^1(\Omega,\cal{F},P)$.