Let $X$ and $Y$ be independent r.v. such that $X \sim N(0,\sigma^{2})$, $Y \sim \Gamma(a,b)$, $Z = \sqrt{X^{2} + Y^{2}}$. Where $\Gamma(a,b)$ denotes the gamma distribution with shape $a$ and scale $b$ and $N$ denotes normal distribution. What is the distribution of $Z$?
It is trivial to show that $X^{2}\sim \Gamma(0.5, 2\sigma^{2})$, Based on the property of generalized gamma distribution $Y^{2}\sim GG(a, 0.5, b^{2})$. Where $GG(a,c,b)$ denotes the generalized gamma distribution with shape $a$ and scale $b$.
I think next applying convolution theorem, the distribution of $X^{2}+Y^{2}$ can be defined.