Suppose we have two sequence of random variables $(X_n)$ and $(Y_n)$, such that
- $Y_n\rightarrow Y \quad \text{in} \quad \mathcal{L}_2,$ and
- $X_n=\mathbb{E}(Y_n \mid X_n),$ for each $n$.
I am wondering if there is a subsequence $(X_{n_k})$ that converges in $\mathcal{L}_2$. I suspect one can construct a counterexample to this statement, but have not been able to do so myself.