What is the probability density function (pdf) of the dot product of M complex normal random variables? What would be the probability density function (pdf) of the complex random variable given below?
$$Z = \sum_{i=1}^{M}{x_{i}^{*}y_{i}}$$
where $x_i, y_i$ are independent r.v.'s with $\mathcal{CN}(0,c)$.
 A: Note that a complex multivariate normal random variable in $\mathbb{C}^n$ is equivalent to a real multivariate normal in $\mathbb{R}^{2n}$.
You have not specified the joint distributions of 1) the real/imaginary components of each variable, 2) the different variables. If the variables are jointly normal, then for $z,w\in\mathbb{C}^n$, with $z=x+iy$ and $w=u+iv$, we have $[x,y,u,v]\sim\mathrm{N}_{0,C}$ for some covariance matrix $C$.
If we assume that $C=I_{4n\times{4n}}$, then all of the components are independent of each other, and some progress can be made.
For the $n=1$ case, the dot product is
$$\bar{z}w=(xu+yv)+i(xv-yu)$$
As the components $x,y,u,v$ are independent and standard normal, this means that 
the dot-product components $\Re[\bar{z}w]$ and $\Im[\bar{z}w]$ will each have a standard Laplace distribution.
For $n>1$ the dot-product components will each be a sum of $n$ independent Laplace-distributed variables. Eventually this will tend to normal, by the central limit theorem. But for finite $n$ the distribution will have no clean expression (e.g. see here), although computing its characteristic function should be straightforward.
