# Relationship between complex normal and bivariate normal distributions

Suppose I have a complex random variable $X$ which follows a complex normal distribution (with $0$ mean). I've been trying to represent the complex normal in a simpler way, but I'm not sure how. Is there a way to represent the complex normal distribution as bivariate normal distribution?

There's a special case where if the mean $\mu$ and the relation matrix $C$ are trivial, then we can use a bivariate normal distribution (by using the covariance matrix $\Gamma$). I'm trying to prove that my complex random variable $X$ follows this special case, where $\mu=E[X]$ and $C=E[(X-\mu)(X-\mu)^{T}]$. However, I'm not sure what does the expectation of a vector random variable represents (is it just the vector of the expectations of the component random variables?). As well, what would $E[(X-\mu)(X-\mu)^{T}]$ represent? (is $(X-\mu)(X-\mu)^{T}$ a vector, matrix or scalar?). Any references related to the subject would be greatly appreciated.

Thanks for the help.

• The definition at en.wikipedia.org/wiki/Complex_normal_distribution starts "Suppose $X$' and $Y$ are random vectors in $\mathbb{R}^k$ such that $\text{vec}[X\, Y]$ is a $2k$-dimensional normal random vector. Then we say that the complex random vector $Z = X + iY$ has the complex normal distribution" and then goes on to state the relationships between the covariance matrices of $X$ and $Y$ and the covariance and relation matrices of $Z$ – Henry Jul 13 '16 at 19:42
• Assuming $X$ is a real random vector, $(X-\mu)(X-\mu)^{T}$ is a real random matrix and $E[(X-\mu)(X-\mu)^{T}]$ is the covariance matrix of $X$ – Henry Jul 13 '16 at 19:45