What is known about the PDFs for the $\ell^2$-norm of these multivariate distributions?

Question

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 on pp. 4-5):

• Kotz
• Student
• Exponential
• Logistic
• Laplace
• Bessel

Thank you.

Note: I asked this question over at MSE, but it received practically no attention. Do let me know if this is off-topic, as this is my first post here.

Answer 1

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.

Answer 2

Actually the paper you cited presented majority of the results that are known currently, a possible supplement is [1]. I will approach this problem by explaining the involvement of spherically symmetric distributions in statistics.

Spherically symmetric distributions are usually used to modeled covariance matrices of spatial processes in spatial statistics. In this regard, you may want to look at [5]. However, due to the increasing concern of non-stationarity of the covariance structure, this method of modeling is out-dated but still useful when the major concern is analysis of local spatial features as well as robustness of the process.

Spherically symmetric distributions are also useful in statistical mechanics[2] and therefore some analysis of classical ensembles. In this regard, you can refer to [3], whose formalism is basically on Hilbert spaces and hence not particularly hard to understand(at least for me...).

In recent years(started from 1980s)[6], another growing branch of statistics that makes intense use of the spherically symmetric distribution is the directional statistics(Or some people call it statistical shape analysis). One example is the Fisher distribution on a sphere; another example is the projected normal distribution [4], which is particularly useful when the data is indeed embedded into a sphere.

Reference

[1]Serfling, Robert J. "Multivariate symmetry and asymmetry." Encyclopedia of statistical sciences (2006).

[2]http://www.df.unipi.it/~konishi/SymmetryStatistics.pdf

[3]Von Neumann, John. Mathematical foundations of quantum mechanics. No. 2. Princeton university press, 1955.

[4]https://stats.stackexchange.com/questions/263896/moment-mgf-of-cosine-of-directional-vectors

[5]Gelfand, Alan E., et al., eds. Handbook of spatial statistics. CRC press, 2010.

[6]Mardia, Kanti V., and Peter E. Jupp. Directional statistics. Vol. 494. John Wiley & Sons, 2009.

[7]Jupp, P. E., and K. V. Mardia. "A unified view of the theory of directional statistics, 1975-1988." International Statistical Review/Revue Internationale de Statistique (1989): 261-294.