Alternating Sequence of Conditional Expectations

Question

Suppose that we have a sequence of random variables ${X_n}, n \in \mathbb{N}_0$. Suppose we have two sub-σ-fields that are not nested, $\mathcal{G}_1$ and $\mathcal{G}_2$. We have some random variable $X$ and define this sequence as,

$X_0 = X$

$X_{k+1} = \mathbb{E}[X_{k}|\mathcal{G}_1]$ if $k$ is even

$X_{k+1} = \mathbb{E}[X_{k}|\mathcal{G}_2]$ if $k$ is odd

For visualization here is the 4th and 5th element of the sequence, $X_4 = \mathbb{E}[\mathbb{E}[\mathbb{E}[\mathbb{E}[X|\mathcal{G}_1]|\mathcal{G}_2]|\mathcal{G}_1]|\mathcal{G}_2]$

$X_5 = \mathbb{E}[\mathbb{E}[\mathbb{E}[\mathbb{E}[\mathbb{E}[X|\mathcal{G}_1]|\mathcal{G}_2]|\mathcal{G}_1]|\mathcal{G}_2]\mathcal{G}_1]$

Is there anything we can say about the convergence of such a sequence? How about required conditions when this converges? I'm quite unfamiliar with martingales and the such, but if at least one has resources that they can recommend for me to understand problems specifically of this type I would be happy as I recently am dealing with quite a few problems of this type.

Answer 1

Answer when $X$ is square integrable: there is a well known theorem in Functional Anlysis (called the von Neumann alternating projections theorem) which says that if $P$ and $Q$ are projections on a Hilbert space with ranges $M$ and $N$ then $PQPQPQ...$ converges to the projection on $M\cap N$. Ref: Hilbert Space Problem Book by P R Halmos, Problem 96. This implies that $X_n$ converges to $E(X|\mathcal G_1 \cap \mathcal G_2)$.

Answer 2

An answer is contained in the following pdf by Diaconis and Khare. In particular, it say that the condition $$ \mathbb E\left[\left\lvert X\right\rvert \log \left(1+\left\lvert X\right\rvert\right)\right]<+\infty $$ is necessary and sufficient for the almost sure convergence of $(X_n)$ to $\mathbb E\left[X\mid \mathcal G_1\cap\mathcal G_2\right]$.