LEt $x$ be a multivariate normal random variable $x\in\mathbb{R}^{n}\sim\mathcal{N}(0,\Sigma)$, where $\Sigma\in\mathbb{R}^{n \times n}$ is a diagonal covariance matrix with entries $[\sigma_1^2,\dots,\sigma_n^2]$ where, in general, $\sigma_1^2\neq\dots\neq\sigma_n^2$.
In the special case of $\sigma_1^2=\dots=\sigma_n^2=1$, the norm of this random variable would be distributed according to a Chi distribution. It is also possible to consider the case $\sigma_1^2=\dots=\sigma_n^2=c$ for some $c > 0$ with appropriate scaling. I was wondering: is it possible to derive a similar distribution for the case in which the diagonal variances are $\sigma_1^2\neq\dots\neq\sigma_n^2$? Can you recommend any resources that might discuss this topic?
(I am aware of this similar question, but the links provided don't seem useful.)
