Assume that $X=X_1 + X_2 +...+X_n$, where $X_i \sim CN(0,\sigma^2)$ and independent. Here $CN$ means circular complex Gaussian.
The question is, what is the distribution for
$Z = \frac{\left|X\right|^2}{\left|X_1\right|^2 + \left|X_2\right|^2+...+\left|X_n\right|^2}$
How can we benefit from the results obtained here: Distribution of the ratio of dependent chi-square random variables
