Suppose $X, Y$ are $L^1$ random variables, and $X\_t$ and $Y\_t$ are real valued stochastic processes with $X\_t, Y\_t \in L^1$ for all $t$ such that the following convergences hold:


i) $X\_t \to X$, $Y\_t \to Y$ a.s and in $L^1$.


ii) $E(X\_t| Y\_t) \to E(X)$ a.s.


iii) $E(Y\_t|X\_t) \to E(Y)$ a.s.


Does it follow that $X$ and $Y$ are independent?



Independence of limits of asymptotically independent processes
The title mentions asymptotic independence, but I don't see any sort of asymptotic independence condition here. For random variables $X$ and $Y$, the conditions $E(X|Y)=E(X)$ a.s. and $E(Y|X)=E(Y)$ a.s. don't imply that $X$ and $Y$ are independent.


For your question, you can take $(X,Y)$ to be uniformly distributed on the eight values $(1,1), (1,-1), (-1,1), (-1,-1), (2,2), (2,-2), (-2,2), (-2,-2)$. Observing $X$ gives you information about the magnitude of $Y$, but not its sign (and vice versa). Then let $X\_t=X$ and $Y\_t=Y$ for all $t$.


