Products of Conditional Expectation Operators: Convergence and Divergence

In this paper, we investigate the convergence of products of conditional expectation operators. We show that if $$(\Omega ,\mathcal {F},P)$$ is a probability space that is not purely atomic, then divergent sequences of products of conditional expectation operators involving 3 or 4 sub-$$\sigma $$-fields of $$\mathcal {F}$$ can be constructed for a large class of random variables in $$L^2(\Omega ,\mathcal {F},P)$$. This settles in the negative a long-open conjecture. On the other hand, we show that if $$(\Omega ,\mathcal {F},P)$$ is a purely atomic probability space, then products of conditional expectation operators involving any finite set of sub-$$\sigma $$-fields of $$\mathcal {F}$$ must converge for all random variables in $$L^1(\Omega ,\mathcal {F},P)$$.