 Characteristic function of a random vector

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 $A$.