Jointly Gaussian Families of Random Variables and Gaussian Processes

Let $(\Omega,\mathcal{F},P)$ be a probability space, let $J$ be a nonempty set, and let $(X_j)_{j\in J}$ be a family of random variables on $(\Omega,\mathcal{F},P)$.

The family $(X_j)_{j\in J}$ is jointly Gaussian (a Gaussian family) if for every natural number $d$ and all distinct indices $j_1,\dots,j_d\in J$, the tuple $(X_{j_1},\dots,X_{j_d})$ is a Gaussian random vector.

For a finite family this agrees with Gaussian Random Vectors and Jointly Gaussian Random Variables: the full tuple is itself such a subfamily, and conversely every subfamily of a Gaussian random vector is a Gaussian random vector by Affine Transformations of Gaussian Random Vectors are Gaussian (reordering being a special case of selecting a subfamily index by index).

When $J$ is a set of real numbers regarded as time indices, so that $(X_t)_{t\in J}$ is a stochastic process, a jointly Gaussian family is also called a Gaussian process.