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

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• DrDu
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.

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 $h(z) = h(1)^z = \exp(z\log h(1))$.

Look up the Maxwell characterization of the multivariate normal distribution.

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