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Let $X$ be a multivariate normal $\mathcal{N}(\mu, \Sigma^2)$ and let $X$ be anistropic, 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 for $X$ 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 a 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 (Generic sum of Gamma random variables) and this seems more complicated to be honest.

On the other hand it feels intuitively to be somewhat similar to a truncated normal distribution but with the truncation happening over a unit ball and this also seems complicated.

kjetil b halvorsen
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rostader
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    It's not a truncated distribution. Its shape will depend on the relative values of *all* the diagonal elements: there's no general simplification. Thus, it's not worth mathematical analysis: in a statistical setting one would seek approximations or use numerical methods. – whuber Jul 24 '21 at 20:39
  • Could you elaborate on why is not worth the analysis. I agree it depends upon the diagonal values. But why is such an analysis not possible – rostader Jul 24 '21 at 21:22
  • Also could you expand on approximations – rostader Jul 24 '21 at 21:38
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    Such an analysis is possible, but the result is necessarily messy because it will depend on every one of the parameters--the $\mu$ and the $\Sigma^2$ components. How insightful or useful would that be? I use "approximation" in its standard sense: formulas that are much simpler and will do the intended job, albeit with some error (that we would hope to provide bounds for). – whuber Jul 25 '21 at 14:30
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    This is a duplicate of a question I answered on MathOverflow: https://mathoverflow.net/questions/398261/norm-contrained-gaussian-distribution/398272#398272 – Matt F. Jul 26 '21 at 04:36

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