 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 $f:[0,\infty )\to \mathbb{R}$.