A Random Vector Is Gaussian with Mean m and Covariance Σ Exactly When Its Law Is N(m,Σ)

In the setting of Gaussian Random Vectors: Standing Notation, let $d\in\mathbb{N}$, $m\in\mathbb{R}^{d}$, let $\Sigma\in\mathcal{S}(d)$ be positive semidefinite, and let $X$ be a random vector in $\mathbb{R}^{d}$.

(i) Gaussian vectors are those with a Gaussian law. $X$ is a Gaussian random vector with $\mathbb{E}X=m$ and $\operatorname{Cov}(X)=\Sigma$ if and only if

$$P_{X}=N(m,\Sigma).$$