I'm looking for resources giving the PDFs for the $\ell^2$-norm of various spherically symmetric, continuous multivariate distributions. 


For instance, the PDF for the $\ell^2$-norm of a multivariate standard normal distribution can be shown to be the chi distribution. I'm looking for similar results regarding any or all of the following multivariate distributions (found [here](http://aurelie.boisbunon.free.fr/downloads/loisSS.pdf) on pp. 4-5):


* Kotz
* Student
* Exponential
* Logistic
* Laplace
* Bessel


Thank you.


*Note: I asked [this question over at MSE](https://math.stackexchange.com/questions/2191166/the-probability-of-a-given-length-of-a-vector-drawn-from-a-multivariate-distribu), but it received practically no attention. Do let me know if this is off-topic, as this is my first post here.*



What is known about the PDFs for the $\ell^2$-norm of these multivariate distributions?
Given a spherically symmetric distribution $P(\vec{x})=f(y)$ of an $n$-dimensional vector $\vec{x}$, of length $y=|\vec{x}|$, then the distribution $P(u)$ of the $\ell^2$-norm $u=|\vec{x}|^2=\sum\_{i=1}^n x\_i^2$ is just


$$P(u)\propto \frac{1}{u} u^{n/2}\,f(\sqrt{u}),$$


omitting a normalization constant. So for the six distributions you list one finds:


* Kotz: $f(y)\propto y^{2N-2}e^{-ry^2/\sigma^2} \Rightarrow P(u)\propto u^{N-2+n/2}e^{-ru/\sigma^2}$
* Student: $f(y)\propto\left(1+\frac{y^2}{\nu\sigma^2}\right)^{-(\nu+n)/2}\Rightarrow P(u)\propto u^{n/2-1}\left(1+\frac{u}{\nu\sigma^2}\right)^{-(\nu+n)/2}$
* Exponential: $f(y)=e^{-\frac{1}{2}y^{2b}/\sigma^{2b}} \Rightarrow P(u)\propto u^{n/2-1}e^{-\frac{1}{2}u^{b}/\sigma^{2b}}$
* Logistic: $f(y)\propto e^{-\frac{1}{2}y^2/\sigma^2}\left(1+e^{-\frac{1}{2}y^2/\sigma^2}\right)^2\Rightarrow P(u)\propto u^{n/2-1}e^{-\frac{1}{2}u/\sigma^2}\left(1+e^{-\frac{1}{2}u/\sigma^2}\right)^2$
* Laplace: $f(y)\propto K\_0(\sqrt{2}y/\sigma)\Rightarrow P(u)\propto u^{n/2-1}K\_0(\sqrt{2u}/\sigma)$
* Bessel: $f(y)\propto y^q I\_q(y/r)\Rightarrow P(u)\propto u^{(n+q)/2-1}I\_0(u^{1/2}/r)$


I have omitted all normalization constants for brevity. Some of these distributions have names, a Gamma distribution for the first and third on your list, a Fisher distribution for the second, no name I know of for the last three.


