Take a probability space $(\Omega,\mathcal{F},\mathbf{P})$ and random variable $X \in L^2(\Omega,\mathcal{F},\mathbf{P})$.

Define the *iterated expectations* of *X* as follows: $X_0 = X$, and, inductively, $X_k = \mathbf{E}[X_{k-1} \, | \, \mathcal{G}_k]$, where $\mathcal{G}_k \subseteq \mathcal{F}$ is some $\sigma$-algebra. Assume that all the $\mathcal{G}_k$ come from some finite set of distinct $\sigma$-algebras. In words, we are repeatedly taking conditional expectations with respect to various information.

**Can we conclude that the sequence $(X_k)$ converges to some limit in $L^2$?**

The paper [1] says the question seems to be open (as of its publication) and reviews some of the literature, going back to Rota's *Alternierende Verfahren*. (The contribution of the paper is to show that if we drop the restriction of finitely many distinct $\sigma$-algebras, the statement is not true.) Has there been any progress since?

[1] M. Akcoglu and J. King. An example of pointwise non-convergence of iterated conditional expectation operators. Israel J. Math. 94 (1996). This paper was pointed out to me by Anthony Quas at a related question -- When do iterated conditional expectations converge?