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 $(\mathrm{\Omega },\mathcal{F},\mathcal{P})$ 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 $L^2(\mathrm{\Omega },\mathcal{F},\mathcal{P})$. This settles in the negative a long-open conjecture. On the other hand, we show that if $(\mathrm{\Omega },\mathcal{F},\mathcal{P})$ 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 $L^1(\mathrm{\Omega },\mathcal{F},\mathcal{P})$.