I Citation needed: Only multivariate rotationally invariant distribution with iid components is a multivariate normal distribution

[DrDu]

TL;DR Summary

I need a citation for the proposition that the only multivariate rotationally invariant distribution with iid components is a multivariate normal distribution.

I need a citation for the following proposition: Assume a random vector $X=(X_1, ..., X_n)^T$ with iid components $X_i$ and mean 0, then the distribution of $X$ is only invariant with respect to orthogonal transformations, if the distribution of the $X_i$ is a normal distribution.
Thank you for your help!

 

[pasmith]

The PDF of X is <br /> f(x_1)\dots f(x_n) where f is the PDF of each X_i. Invariance under orthogonal transformations would require f to be even, since the transformation which multiplies the ith component by -1 and fixes the others is orthogonal. We can then write f(z) = g(z^2) whilst g(x_1^2) \cdots g(x_n^2) = F(x^TAx) for some symmetric matrix A which satisfies R^TAR = A for every orthogonal R. This is equivalent to the requiement that A should commute with every orthogonal R. I believe this in fact results in A being a multiple of the identity. If so, we have <br /> g(x_1^2) \cdots g(x_n^2) = F(x_1^2 + \dots + x_n^2) where the multiplier of the identity has been absorbed into F. Setting all but one of the x_i to be zero then shows that <br /> g(x_j^2)g(0)^{n-1} = F(x_j^2). Setting g = Ch where h(0) = 1 we find <br /> F = C^n h where <br /> h(z_1) \cdots h(z_n) = h(z_1 + \dots + z_n) for all (z_1, \dots, z_n) \in [0, \infty)^n. I think now we can proceed by induction on n, noting that for n = 2 and the assumption of continuous h we have h(z) = h(1)^z = \exp(z\log h(1)).

 

[statdad]

Look up the Maxwell characterization of the multivariate normal distribution.

 

[DrDu]

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On a Characterization of the Normal Distribution​

M. Kac
American Journal of Mathematics, Vol. 61, No. 3 (Jul., 1939), pp. 726-728 (3 pages)
https://doi.org/10.2307/2371328•https://www.jstor.org/stable/2371328


Thank you guys, your comments helped me to locate the article mentioned above!