Given the following function of random variables


$$g = \frac{1}{n} \sum\_{k=1}^{n}{|h\_k|\exp\left( j \theta\_k \right)},$$
where $h\_1, \cdots, h\_n$ are i.i.d. random variables following the complex Gaussian distribution $\mathcal{CN}(0,\beta)$ and $\theta\_1, \cdots, \theta\_n$ are i.i.d. random variables with probability density function (PDF) given by $\frac{1}{2\pi}$ ($i.e.$, the uniform distribution). Additionally, we assume that $h\_k$ and $\theta\_k$ are independent for all values of $k$. 


What would be the PDF of $g$ for small $n$ ($e.g.$, n = 2) and for the case where $n \gg 1$?


This problem arises from the study on wireless communications channels and is of great importance to the research community.



PDF of $g = \frac{1}{n} \sum_{k=1}^{n}{|h_k|\exp\left( j \theta_k \right)}$?
For the complex Gaussian distribution the real and imaginary parts of $h\_k$ are i.i.d. with a normal distribution; the absolute value $|h\_k|$ has distribution $P(|h\_k|)=|h\_k|\exp(-|h\_k|^2/2)$ and the argument $\phi\_k={\rm arg}\,h\_k$ is uniformly distributed in $(0,2\pi)$, independently of $|h\_k|$. So to generate the random variable $G\_k=|h\_k|e^{i\phi\_k}$ you may either draw a real random number $|h\_k|$ with distribution $P(|h\_k|)$ and a second independent random number $\phi\_k$ uniformly in $(0,2\pi)$, or equivalently draw the complex variable $G\_k$ directly from a complex Gaussian. 


Then $g$ is a sum of independent complex Gaussians $G\_k$, which is again a complex Gaussian.


