I'm trying to understand the difference between Elliptic and Rotationally invariant distributions.
Previously, for some reason, I was under the impression that the Gaussian distribution was the only rotationally invariant distribution, i.e. it is the only distribution with the property that $$f(\mathbf{x}) = f(\Vert \mathbf{x} \Vert) =f(\Vert R \mathbf{x} \Vert)=f(R\mathbf{x}),$$ where $\mathbf{x} \in \mathbb{R}^n$ and $R$ is an $n \times n$ orthogonal matrix, since $\Vert \mathbf{x} \Vert = \Vert R \mathbf{x} \Vert$.
Looking at the wikipedia article on Elliptic distributions, assuming $0$ mean and diagonal covariance matrix their pdfs have the form $$f(x) = k \cdot g( \Vert \mathbf{x} \Vert ),$$ where $k$ is some scaling factor.
Doesn't this mean that elliptic distributions with $0$ mean and diagonal covariance matrix ($0$ centered spherical distributions), are rotationally invariant? Is it true that the Gaussian distribution is not the only rotationally invariant distribution?
