Gaussian Random Vectors and Jointly Gaussian Random Variables

Let $(\Omega,\mathcal{F},P)$ be a probability space and let $d$ be a natural number.

Random variables $X_1,\dots,X_d$ on $(\Omega,\mathcal{F},P)$ are jointly Gaussian, and the tuple $(X_1,\dots,X_d)$ is called a Gaussian random vector, if there exist $m$, either zero or a natural number, independent standard normal random variables $Z_1,\dots,Z_m$ on $(\Omega,\mathcal{F},P)$, and real numbers $\mu_i$ and $a_{ij}$ ($1\le i\le d$, $1\le j\le m$) such that

for $m=0$ the family $Z_1,\dots,Z_m$ is empty and the sum is the empty sum, equal to $0$.

The data $\bigl(m,(\mu_i),(a_{ij}),(Z_j)\bigr)$ is called a Gaussian representation of $(X_1,\dots,X_d)$. A single random variable $X$ is called Gaussian (or normally distributed) if $(X)$ is a Gaussian random vector with $d=1$. Constant random variables are Gaussian (take $m=0$), and every standard normal random variable is Gaussian (take $d=m=1$, $\mu_1=0$, $a_{11}=1$); degenerate Gaussian random variables are thus allowed.