Orthogonally invariant random vector

A random vector $X\in {\mathbb{R}}^d$ is orthogonally invariant when $AX$ has the same distribution as $X$ for every $A\in O(d)$. Its characteristic function is radial and real: ${\phi }_X(u)=f(\Vert u{\Vert }^2)$ for a continuous function $f:[0,\infty )\to \mathbb{R}$.

If an orthogonally invariant random vector in dimension at least two has independent coordinates, then it is centered isotropic Gaussian, possibly degenerate at zero. Independence and rotational invariance givewhile continuity and $f(0)=1$ imply $f(r)=e^{-\alpha r}$ for some $\alpha \ge 0$. Thus $X\sim N(0,2\alpha I_d)$.