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enter image description hereIn the proving the above, I see that since $X$ is multivariate gaussian then for any orthogonal matrix $Q$ we have that $QX$ is standard multivariate gaussian. Then I somehow reasoned that $Y=X/\|X\|_2$ is also rotational invariant and has norm 1. This Y should be uniformly distributed on a sphere but I cannot write down exactly why this should be true. Can someone give me a rigorous mathematical argument to prove this claim? Thanks

user3503589
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    What's your definition of the uniform distribution on the sphere? – Dominik Jan 06 '16 at 13:46
  • @Dominik I am not sure how would I define a uniform distribution on an n dimensional sphere but it is an extension of uniformly distributed points on a surface 3-dimensional sphere $x^2+y^2+z^2=1$. I just added a figure I found on cross validated – user3503589 Jan 06 '16 at 13:54
  • Use polar coordinates: $\exp(-|x|^2/2),dx=\exp(-r^2/2) r^{n-1},dr,d\sigma$, where $\sigma$ is surface area on the unit sphere in $\Bbb R^n$. – John Dawkins Jan 06 '16 at 14:16
  • Well the problem is that we can't provide a rigorous proof without a rigorous definition of the uniform distribution. – Dominik Jan 06 '16 at 14:18
  • https://math.stackexchange.com/q/444700/321264, https://math.stackexchange.com/q/1864519/321264 – StubbornAtom Jun 01 '20 at 07:43

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I will fix the dimension $n$ and use $S:=\{x\in\Bbb R^n:\|x\|=1\}$ to denote the unit sphere. Let $\sigma$ denote surface measure on $S$, and define $\bar\sigma:=[\sigma(S)]^{-1}\sigma$, the "uniform distribution" on $S$.

Let $f:S\to\Bbb R$ be bounded and Borel measurable. Then $$ \eqalign{ \Bbb E[f(Y)] &=\int_{\Bbb R^n} f(x/\|x\|)(2\pi)^{-n/2}e^{-\|x\|^2/2}\,dx\cr &=(2\pi)^{-n/2}\sigma(S)\int_S\int_0^\infty f(u)e^{-r^2}r^{n-1}\,dr\,\bar\sigma(du),\cr &=\int_S f(u)\,\bar\sigma(du),\cr } $$ because $$ \int_0^\infty e^{-r^2}r^{n-1}\,dr =2^{n/2-1}\int_0^\infty e^{-t}t^{n/2-1}\,dt=2^{n/2-1}\Gamma(n/2) $$ while $\sigma(S)=2\pi^{n/2}/\Gamma(n/2)$.

John Dawkins
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  • Could you please explain how did you get the second term while computing $E[f(Y)]$ Also isn't the surface area of an $n-1$-dimensional sphere $(2\pi)^{n/2}/ \Gamma{(n/2)}$ https://en.wikipedia.org/wiki/N-sphere – user3503589 Jan 10 '16 at 10:32