Are there any general results that say when the expected value of a product of two independent random variables about their means is zero or reduces to a product of central moments?
For instance, let $X$ and $Y$ be independent random variables. We have the following central moments: \begin{align*} E\left[(X - E[X])(Y - E[Y]) \right] &= \text{Cov}(X,Y) \\ \\ E\left[(X - E[X])^2(Y - E[Y]) \right] &= E\left[(X - E[X])(Y - E[Y])^2 \right] = 0 \\ \\ E\left[(X - E[X])^2(Y - E[Y])^2 \right] &= \sigma_X^2 \sigma_Y^2 \end{align*}
Can we generalize what $E\left[(X - E[X])^i (Y - E[Y])^j \right]$ will be?
My intuition is that if one of $i$ or $j$ is odd and the other is even, then $E\left[(X - E[X])^i(Y - E[Y])^j \right] = 0$, but if both are even then $E\left[(X - E[X])^i(Y - E[Y])^j \right] = \mu_{X,i} \mu_{Y,j}$, where $\mu_{X,i}$ and $\mu_{Y,j}$ are the $i$th and $j$th central moments of $X$ and $Y$, respectively.