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In this proof we're trying to show if $X$ is uniform on the sphere, then it is subgaussian. This is done by considering a random vector in R^n and normalising it onto the sphere, by using a multivariate normal vector $g$. Is there any way to do this for the random vector $X$ onto the closed ball instead of sphere?

My attempt:

I thought about taking such a g, normalising it, then multiplying by a uniform random variable on $[0,1]$, but I believe this doesn't work, since the volume of a ball is proportional to the radius to the nth power this fails ( the distribution would be denser towards the centre).

I also thought about using the inverse tan function on the norm of the vector but I don't think that would yield a uniform distribution.

Bernard
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kam
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    Instead of a uniform distribution you want a suitably biased distribution. Two examples are given in the question here: https://math.stackexchange.com/questions/2275465/pdf-of-uniform-distribution-over-the-hypersphere-and-the-hyperball – David K Feb 15 '20 at 18:38
  • By the way, it seems to me that $\langle X,x\rangle$ has support only on a finite interval, which I would have thought would be enough to make it sub-Gaussian. I feel I'm missing the motivation for this particular proof of this particular theorem. – David K Feb 15 '20 at 18:41

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