Characteristic function of a random vector (${\phi }_X(u)=\mathbb{E}[e^{i⟨u,X⟩}]$)

For an ${\mathbb{R}}^d$-valued random vector, the characteristic function is ${\phi }_X(u)=\mathbb{E}[e^{i⟨u,X⟩}]$. It determines the joint distribution and satisfies ${\phi }_{AX}(u)={\phi }_X(A^{T}u)$ for every deterministic matrix $A$.

If $X\sim N(0,{\sigma }^2{I}_d)$, then

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 $X\sim N(0,2\alpha I_d)$.