I'm looking for a reference and a proof of the following version (or eventually a more general version) of the Central Limit Theorem for complex random variables.


**Theorem.** *Let $Z\_1, Z\_2, \dots, Z\_n$ be independent identically distributed complex Goodman (see Note 1) random variables with mean $\mu$ and complex variance $\sigma^2$ (see Note 2). Then as $n \to \infty$
$$(\star)\quad\frac{\sum\_{i=1}^n Z\_i - \mu n}{(\sigma^2 n)^{1/2}}$$
converges in law to a standard complex normal random variable.*


**Note 1.** A complex random variable $Z = X + iY$ ($X, Y \in \mathbb{R}$) is *Goodman* if the covariance of $X$ and $Y$ is zero.


**Note 2.** The complex variance of a complex random variable $Z$ is defined as $\mathbf{E}\left[(Z - \mathbf{E}[Z])\overline{(Z - \mathbf{E}[Z])}\right]$.


The only references I was able to find are:


1. Theorem 7.3, Jonathan Shuster, Properties of the Gaussian Gaussian Distribution (this is an apparently unpublished B.Sc. thesis).
2. Exercise 2.2.14, Terrence Tao, Topics in Random Matrix Theory (this is an exercise without solution).


I would like a published reference, possibly a book, containing a proof.
In general, I'm a bit surprise of not finding books devoted to complex random variables. Perhaps this is due to the fact that many theorems on complex random variables follow easily from theorems on real random variables.


Furthermore, I don't understand why the condition on begin Goodman is needed ((1) provides a counter-example but I don't find it enlightening). Since ($\star$) involves only additive properties on the complex random variables, why one cannot deduce the Central Limit Theorem for complex random variables just be identifying $Z$ with the real vector $(X, Y)$ and using the Central Limit Theorem for vectors?


Thank you for the help.



Reference request and clarification for Central Limit Theorem for complex random variables
On the "Furthermore":


The central limit theorem for vectors involves the variance-covariance matrix. Let $Z = X+iY$ be a complex random variable (with mean $0$ for simplicity). If $X,Y$ have variance-covariance matrix:


$$\begin{pmatrix} \mathbf E[X^2] & \mathbf E[XY] \\ \mathbf E[XY] & \mathbf E [Y^2] \end{pmatrix} = \begin{pmatrix} a & b \\ b & c \end{pmatrix} $$


then the complex variance is $$\mathbb E [ (X+iY) (X- i Y) ]= \mathbb E[ X^2+Y^2] =a+c$$ so knowing the complex variance does not determine the variance-covariance matrix and thus doesn't determine the limiting distribution.


Your definition of Goodman is equivalent to $b=0$. Even under $b=0$ knowing $a+c$ does not determine the full matrix. The reference by Shuster gives a different definition of Goodman, which is equivalent to $b=0$ and $a=c$. It is clear from the matrix perspective why such an assumption is necessary (since every symmetric positive semidefinite matrix can in fact appear). (The reference by Tao doesn't use the notion of Goodman random variables, but has a similar assumption.)


