what is a standard complex gaussian random variable?

Question

I'm reading a book about Gaussian analytic functions where they talk e.g. about the complex gaussian distribution with mean 0 and variance 1. The variance is $E(XX^*)$ for $X$ a variable with this law. Does this mean that if $A$ is a REAL standard Gaussian variables, $A, iA, (\frac{1}{2}+i\frac{\sqrt{3}}{2})A, ...$ are not random complex gausian variables? Otherwise they would have the same parameters without having the same law. (Actually in the book they say that the covariance of a complex gaussian vector characterizes the law, so I'm a bit lost here).

Answer 1

The distribution of a complex Guassian random variable $X$ is not determined by just the mean $EX$ and variance $EX\overset {-} X$. Think of complex random variables as $\mathbb R^{2}$ valued random vectors. You need a mean vector and a $2 \times 2$ variance -covariance matrix to completely determine the Gaussian distribution.