What is the distribution of functions of Nakagami?

Question

Assume we have $U_i$ Nakagami distributed with parameter $m$, for $i\in [1,n]$.

What would the distribution of the following

$$ \big|\sum_i {a_i U_i}\big|^2$$ where $a_i$ are non-negative constants.

Also I know that the distribution of $$|U_i|^2 $$ would be normalized Gamma random variable, is that true? If yes then what is the parameter in this case?

What would be the distribution of the following two functions $$\sum_{i=1}^n|U_i|^2$$ $$\sum_{i=1}^n a_i|U_i|^2$$

if $a_i$ are non-negative constants.

Thanks alot

Answer 1

Partial answer:

If $U_i$ is Nakagami distributed with parameters $(m,1)$, then it means $U_i^2$ is a gamma distribution with parameters $(m,\dfrac{1}{m})$,

By the summation property of Gamma distribution, we have $\sum_{i=1}^n U_i^2$ is a gamma distribution with parameters $(\dfrac{n}{m}, \dfrac{1}{m})$