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Let $X_1,...,X_n$ be $p$-vectors uniformly distributed in the ball $B_r=\{x:\Vert{x}\Vert_2\le r,r\gt 0\}.$

Can someone explain to me what it means by "uniformly distributed in the ball"?

For example, what does $X_i$'s distribution look like? I am having trouble understanding these high-dimensional stuff... .

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  • $\begingroup$ In this thread Ray Koopman suggested a method how to generate multidimensional random data with any kurtosis from normal's to uniform (flat). $\endgroup$
    – ttnphns
    Apr 23, 2017 at 18:22

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I think this should help:

Starting off with a 2D "ball", i.e. a circle. Points are uniformly distributed within a circle.

http://blogs.sas.com/content/iml/2016/03/30/generate-uniform-2d-ball.html

This can then be extended to a 3D "ball", i.e. a sphere and onto higher dimensions.

http://www.statsblogs.com/2016/04/06/generate-points-uniformly-inside-a-d-dimensional-ball/

(I would have posted this as a comment, but I'm 9 short!)

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  • $\begingroup$ Can you extend this answer by summarizing what can be found behind the links? That makes the anser more useful. As it is now it is short and could be closed. $\endgroup$ Apr 23, 2017 at 18:11
  • $\begingroup$ Apologies. I wanted to just share the links as a comment, which I believed would be helpful. But I wasn't able to. $\endgroup$ Apr 23, 2017 at 18:14
  • $\begingroup$ The second link above doesn't work. I have found this one here, which at some point will also stop working... $\endgroup$
    – ngiann
    Jul 20, 2021 at 10:05

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