The following question arose while I was trying to explore possible further extensions of a CLT by Liverani which I mentioned here already (see this link, I can tell you more details upon request). It seems to be of a very technical nature but an answer is important to connect Liverani's (extended) theorem with very recent results on limit theorems.

So, let $T:\Omega\to \Omega$ be an ergodic, invertible transformation on a probability space $(\Omega,\mathcal{F},P)$ and let $\mathcal{F}_{0}\subset \mathcal{F}$ be a sigma field satisfying $\mathcal{F}_{0}\subset T^{-1}\mathcal{F_{0}}:=\{T^{-1}A:A\in\mathcal{F}_{0}\}$, so that $(\mathcal{F}_{n})_{n\in\mathbb{Z}}$ given by $\mathcal{F}_{n}:=T^{-n}\mathcal{F}_{0}$ defines an increasing filtration of sigma fields contained in $\mathcal{F}$.

Denote by $E_{0}:L^{2}_{P}\to L^{2}_{P}$ the conditional expectation with respect to $\mathcal{F}_{0}$ and (also by) $T:L^{2}_{P}\to L^{2}_{P}$ the Koopman operator $TX:=X\circ T$. By $E_{0}T^{k}$ we denote the composition of the operators $E_{0}$ and $T^{k}$ (where, of course, $T^{k}$ is $T$ composed with itself $k$ times). The question is the following: do you know an example of a $\mathcal{F}_{0}-$measurable, centered, function $X_{0}\in L^{2}_{P}$ for which the series \begin{equation} \sum_{n\geq 0}X_{0}E_{0}T^{n}X_{0} \end{equation} converges almost surely but the series \begin{equation} \sum_{n\geq 0}E_{0}T^{n}X_{0} \end{equation} diverges in a nonnegligible set? If not, can you prove that the a.s convergence of the first series implies the a.s convergence of the second one? (note that the reciprocal implication is trivial, and that the problem arises dealing with the set $[X_{0}=0]$.)