Let $X$ be a multivariate normal $\mathcal{N}(\mu, \Sigma^2)$ and let $X$ be anisotropic, that is I am considering $\Sigma$ to be a diagonal matrix but the elements on the diagonal might be different.
I am interested in finding the distribution of $X/\|X\|_2$.
As a start let $X$ be isotropic. Then $\|X\|_2^2$ will be Gamma distributed. But then $\|X\|_2$ will perhaps have to follow what I found to be a Nakagami distribution (https://en.wikipedia.org/wiki/Nakagami_distribution). So I need to find the ratio of a normal and this Nakagami distribution. However for the anisotropic case $\|X\|_2^2$ will not be Gamma but a mixture of Gamma distributions (https://stats.stackexchange.com/questions/72479/generic-sum-of-gamma-random-variables) and this seems more complicated to be honest.
On the other hand it feels intuitively somewhat similar to a truncated normal distribution, but with the truncation happening over a unit ball and this also seems complicated.
Any ideas/ references/ hints/ thoughts
Thanks