Mean Independence and uncorrelated vs independence

Question

If we are given 2 RV's X and Y and find that they are both mean independent and uncorrelated, is it suffice to say that they are independent or are they any situations that this is not true? (Given that their expectations exist)

Answer 1

The standard counterexample for "uncorrelated normals are not necessarily independent" works. Let $X$ be $N(0,1)$ and $Z$ be an independent coinflip $\pm 1,$ and let $Y=XZ.$ $X$ and $Y$ are mean-independent: $$E(X\mid Y) = E(Y\mid X) = E(X)=E(Y)=0.$$ (For instance $E(Y\mid X) = XP(Z=1)+(-X)P(Z=-1) = 0.)$

But they are not independent, since, e.g. $E(|Y|\mid X) = X$ which is generally not equal to $E(|Y|).$