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 $$f(x_1)\dots f(x_n)$$ 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 $i$th 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 $$g(x_1^2) \cdots g(x_n^2) = F(x_1^2 + \dots + x_n^2)$$ 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 $$g(x_j^2)g(0)^{n-1} = F(x_j^2).$$ Setting $g = Ch$ where $h(0) = 1$ we find $F = C^n h$ where $$h(z_1) \cdots h(z_n) = h(z_1 + \dots + z_n)$$ 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!