[Edited: Since my previous question is not so clear, I changed my question.]
Let $X,Y$ be random variables in $\mathbb{R}$ such that $$ \mathbb{E}[XY] = 0.$$ If $\mathbb{E}[X] = \mathbb{E}[Y] = 0$, $X$ and $Y$ are uncorrelated. (However, if possible, I don't want to make zero mean assumption.)
Here is my question: Based on the above information $\mathbb{E}[XY] = 0$, what other conditions are needed if one can conclude
If $\mathbb{E}[Y]=0$, $X^i$ and $Y$ are uncorrelated. Or without zero mean, $$ (*) \quad \mathbb{E}[X^iY]=0.$$
For example, if $X$ and $Y$ are independent with $\mathbb{E}[Y]=0$, $(*)$ satisfies. But this is too trivial. And I am trying to find other conditions which makes $(*)$ true (even for $i=2$ or $3$). I am thinking about some conditions on the distributions of $X$ or $Y$.
Any comments or suggestions will be very appreciated. Thanks in advance.
p.s. I am sorry for changing my question.