Maxwell characterization of the normal distribution
= Maxwell characterization of the normal distribution
{c}
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 give
$$
f(r+s)=f(r)f(s),
$$
while continuity and $f(0)=1$ imply $f(r)=e^{-\alpha r}$ for some $\alpha\geq0$. Thus $X\sim N(0,2\alpha I_d)$.