If a multivariate distribution is radially symmetric and its marginals are mutually independent, then it must be normal.
Does the same hold if the marginals are merely pairwise independent?
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Neat Theorem to start off with. Do you have a reference for it? – muaddib Jun 23 '15 at 00:42
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1No I don't, sorry. It's pretty quick to establish in the 2D case by setting $f(x) f(y) = g(x^2 + y^2)$, switching to polar coordinates, differentiating with respect to $\theta$, and rearranging into the ODE for the Gaussian. Unless I'm missing something silly, essentially the same argument holds in higher dimensions (by switching to cylindrical coordinates) to show that $f$ satisfies $f'(x) \propto x f(x)$. I'm pretty sure it's direct and what I said above suffices, but I'm also terrified of making a mistake in front of the entire internet, so perhaps I should edit my question to reflect that. – user3047059 Jun 23 '15 at 01:55
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Here we go, I found one. See the first sentence of the abstract: http://www.orga.cvss.cc/media/publications/SinzGerwinnBethge_2008.pdf – user3047059 Jun 24 '15 at 01:25
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Thank you so much for following up with a reference. Cool to see this is due to Kac. I've previously found myself enjoying his work again and again. Cheers. – muaddib Jun 25 '15 at 03:00