Suppose that we have distributions $F\_1 $ and $F\_2$. Under what conditions on $F\_1,F\_2$ is it possible to construct random variables $X\sim F\_1,Y\sim F\_2$ such that $Y=E(X|\mathscr{G})$, that is, $Y$ is the conditional expectation of $X$ with respect to some $\sigma$-field $\mathscr{G}$?



If $X∼F_1$, $Y∼F_2$, under what conditions on $F_1$, $F_2$ can we construct $Y=E(X\mid\mathscr{G})$ for some $\mathscr{G}$?
The condition proposed by Nate Eldredge is not only necessary, but also sufficient. This is classical Strassen's theorem:



> 
> **Theorem** For integrable random variables $X$ and $Y$ the following is equivalent:
> 
> 
> * There exist $\hat X \overset{d}{=} X$ and $\hat Y \overset{d}{=} Y$ such that $E[\hat X | \hat Y] = \hat Y$.
> * $Y\prec X$ in convex stochastic order, i.e. $E[f(Y)]\le E[f(X)]$ for any convex function $f$.
> 
> 
> 


Note that the formulation in [**Strassen's paper**](http://projecteuclid.org/download/pdf_1/euclid.aoms/1177700153) differs from this one (and Strassen attributed it himself to Hardy-Littlewood-Polya-Blackwell-Stein-Sherman-Cartier-Fell-Meyer). In the present form it is given e.g. in [**Ruschendorf**](http://projecteuclid.org/download/pdf_1/euclid.aop/1176994468) (Theorem 6 for $i=2$).


