# When do iterated conditional expectations converge?

Take a probability space $$(\Omega,\mathcal{F},\mathbf{P})$$ and random variable $$X$$ satisfying $$\mathbf{E}[|X|]<\infty$$.

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 $$\sigma$$-algebras.

In words, we are repeatedly taking conditional expectations with respect to various information.

Can we conclude that the sequence $$(X_k)_k$$ converges $$\mathbf{P}$$-almost surely?

In my comment on the question I include some of what I know about this, but I suspect that this is easy if one looks at it right, or covered in a standard reference.

• When $\Omega$ is finite, tau.ac.il/~samet/papers/iterated.pdf proves the assertion by viewing $\mathbf{E}[\cdot \, | \, \mathcal{G}_i ]$ as a Markov transition matrix (Samet's Proposition 2). Maybe a general versioin can obtained by extending this approach, but hairy uncountable-space Markov processes seem like overkill - the assertion feels more elementary to me. I was tempted to find a reversed martingale related to the sequence $(X_k)$, but have had no success with this approach -- since the $\mathcal{G}_i$ are not nested, it is not obvious how to do this. – Ben Golub Feb 9 '14 at 19:18
• @StephanSturm -- thanks! I think you mean if they are nested. But since it convergence is known for finite probability spaces, irrespective of nestedness, I'd be surprised if it became a problem only once we go to general probability spaces. – Ben Golub Feb 9 '14 at 22:21
• @StephanSturm Re the "finite set", I just mean that only finitely many different $\mathcal{G}_k$ appear -- e.g. first we condition on $\mathcal{G}_1$, then on $\mathcal{G}_2$, then on $\mathcal{G}_1$ again, etc. I am not sure if this restriction is important. – Ben Golub Feb 9 '14 at 22:26
• I'm pretty sure it's false, and think I've seen a reference showing this, but don't remember it now - will try to reconstruct. – Anthony Quas Feb 9 '14 at 23:17
• @aduh a little, in terms of understanding what's known: see answer below – Ben Golub Nov 7 '18 at 14:10

M. Akcoglu and J. King. An example of pointwise non-convergence of iterated conditional expectation operators. Israel J. Math. 94 (1996)

• thanks so much, this is very helpful. Do you happen to know whether there has been any progress on the question of L^2 convergence when there are only finitely many conditioning sigma-algebras? It's described as an open question in the introduction of the paper you cite. – Ben Golub Feb 10 '14 at 3:29
• Sorry. I don't know. What I would do, though is if you have access to MathSciNet, you can search for papers citing the paper I mentioned. This is somewhat limited because MSN only started tracking citations sometime after 2000, but there's a reasonable chance you'd find anything that had done something. – Anthony Quas Feb 10 '14 at 6:14

Here's what's known, courtesy of Omer Tamuz.

Here's what's known, with references below.

Amemia and Ando (1965) prove weak convergence in $$L^2$$. This also covers the finite-$$\Omega$$ case. Convergence in norm in $$L^2$$ seems to still be open. It is known to converge if your sequence of algebras is periodic, as is shown by Delyon and Delyon (1999). If you more generally allow projections to a closed subspaces of $$L^2$$, then there is a recent example by Kopecká, E. & Paszkiewicz (2017) where no convergence in norm occurs. But not every projection to a subspace corresponds to conditional expectation according to some sub-sigma-algebra.

Amemiya and T. Ando, Convergence of random products of contractions in Hilbert space, Acta Sci. Math. (Szeged) 26 (1965), 239–244.

B. Delyon and F. Delyon, Generalization of von Neumann’s spectral sets and integral representation of operators, Bull. Soc. Math. Fr. 127 (1999), 25–41.

Kopecká, E. & Paszkiewicz, A, Strange products of projections, Israel Journal of Mathematics, April 2017, Volume 219, Issue 1, pp 271–286