The Law of a Gaussian Random Vector Is Determined by Its Mean and Covariance Matrix

In the setting of Probability Spaces: Notation and Product Measures, Lebesgue Measure on ℝⁿ and Image Measures: Notation, let $d\in\mathbb{N}$.

(i) Canonical representation. Let $X$ be a Gaussian random vector in $\mathbb{R}^{d}$, let $m=\mathbb{E}X$ and $\Sigma=\operatorname{Cov}(X)$ (Square-Integrability, Mean and Covariance Matrix of a Gaussian Random Vector (i)), a positive semidefinite matrix (Rules for Expectations, Variances and Covariances (xii)). If $\Sigma=0_{d}$, then $X=m$ almost surely. If $r\in\mathbb{N}$ and $B\in\mathbb{R}^{d\times r}$ are such that $B^{\top}B$ is invertible and $BB^{\top}=\Sigma$ (such $r$ and $B$ exist when $\Sigma\neq0_{d}$, by A Nonzero Positive Semidefinite Matrix Factors as BBᵀ with BᵀB Invertible (i)), then

$$W=(B^{\top}B)^{-1}B^{\top}(X-m)$$

is a standard normal vector in $\mathbb{R}^{r}$, and

$$X=m+BW\quad\text{almost surely}.$$

(ii) Mean and covariance determine the law. Let $X$ be a Gaussian random vector in $\mathbb{R}^{d}$, let $(\Omega',\mathcal{F}',P')$ be a probability space and let $X'$ be a Gaussian random vector in $\mathbb{R}^{d}$ on it; both are square-integrable (Square-Integrability, Mean and Covariance Matrix of a Gaussian Random Vector (i)). If $X$ and $X'$ have the same mean and the same covariance matrix, each computed on its own probability space, then $X$ and $X'$ are identically distributed.

(iii) Every mean and covariance occurs. For every $m\in\mathbb{R}^{d}$ and every positive semidefinite $\Sigma\in\mathcal{S}(d)$, there are a probability space $(\Omega',\mathcal{F}',P')$ and a Gaussian random vector $X'$ in $\mathbb{R}^{d}$ on it with

$$\mathbb{E}X'=m,\qquad\operatorname{Cov}(X')=\Sigma.$$