Condition for complex Gaussian RVs We know that a real-valued Gaussian RV is completely speciﬁed by its mean and variance. Can we say the same thing for a complex-valued Gaussian RV?
If we consider a complex-valued RV Z, defined as Z = X + j Y, where X and Y are real-valued RVs, for Z to be Gaussian, is it necessary that X and Y are both Gaussian RVs? Or at least jointly Gaussian?
 A: It depends on what meaning you assign to a complex RV "being Gaussian."
If you say that $Z$ is Gaussian iff $$f(z) = k e^{-\frac{z^2}{2\sigma^2}}$$
then $Z = X+iY$ with $X$ and $Y$ Gaussian reals is not a complex Gaussian.  But that 
definition of complex Gaussian makes very little sense anyway; the integral over the complex plane of that distribution diverges.
If instead you define a complex Gaussian as 
$$f(z) = k e^{-\frac{\overline{z}z}{2\sigma^2}}$$
then yes it is necessary that the $X$ and $Y$ components each be Gaussian.
($k$ is whatever constant is needed to make the integral over the complex plane of $f(z)$ come out to $1$.)
Finally, say you define, for any given line through the origin $L_\theta$, a generalized complex conjugate operator $*_L$ consisting of reflection through the line $L_\theta$ making an angle $\theta$ with the $x$ axis..
For example, $L_0$ is just the $x$ axis itself, and $z^{*_{L_0}} \equiv \overline{z}$.
Then for any $L_{\theta}$, you define the generalized complex Gaussian distribution as 
$$
f(z) = k e^{-\frac{z^{*_{L_\theta}} z}{2\sigma^2}}
$$
it is indeed true that the individual $X$ and $Y$ variables will be jointly Gaussian.
