Suppose $X$, $Y$, $Z$ are zero-mean pairwise uncorrelated random variables with variance $\sigma^2$. If we know that $E(XYZ)$ exists and it is a finite number, then how great can it be? I mean, is there any upper bound?
I tried to use Cauchy–Schwarz inequality and got the following:
$|E(XYZ)|^2 \leq E(X^2) \cdot E(Y^2Z^2) = \sigma^2 E(Y^2Z^2)$
However, how can I find $E(Y^2Z^2)$ ? As far as I understand, if $Y$ and $Z$ are uncorrelated, it doesn't imply that $Y^2$ and $Z^2$ are also uncorrelated.
Any help or hints about possible solution would be greatly appreciated.