does uncorrelation extend to product of complex random variables?

Question

Give two uncorrelated complex variables, $X$ and $Y$. Are $XX^{*}$ and $YY^{*}$ also uncorrelated, where $*$ means complex conjugation?

Answer 1

It depends on distributions:

$$ Cov(|X|^2 , |Y|^2) = E[|X|^2 |Y|^2] - E[|X|^2]E[ |Y|^2]$$

$$ E[|X|^2 |Y|^2] = \int_{x,y}|X|^2 |Y|^2\ f_{X,Y}(x,y) dx dy $$

Only, in case of Gaussian distribution we can say $$ f_{X,Y}(x,y) = f_X(x)f_Y(y) \ \ \ (1)$$

and therefor the co-variance is zero; they are uncorrelated

but, always (1) is true for independent random variables

so, in general the answer depends on distribution of X and Y