Let $P$ be the joint distribution of two random variables $X$ and $Y$, that both have support on $(0,1)$ (I am also interested in the case where $X$ takes values on $k$-dimensional simplex, but I would be happy to start with the simple case).

Now, suppose that for all $n,m \in \mathbb{N}$, we have:

$E[X^nY^m] = E[X^n]E[Y^m]$

Is this a sufficient condition for independence of $X$ and $Y$? Are there other conditions I would need?

I suspect that I may need some knowledge of higher order moment problems: does anyone have any suggestions for useful references?