Independent Jointly Gaussian Families are Jointly Gaussian

Let $(\Omega,\mathcal{F},P)$ be a probability space, let $n\ge1$ be a natural number, and for each $q\in\{1,\dots,n\}$ let $J_q$ be a nonempty set and let $(X^{q}_{j})_{j\in J_q}$ be a jointly Gaussian family of random variables on $(\Omega,\mathcal{F},P)$. Suppose that the generated $\sigma$-algebras

are independent.

Then the combined family consisting of all the random variables $X^{q}_{j}$ ($1\le q\le n$, $j\in J_q$), indexed by the set of pairs $(q,j)$ with $j\in J_q$, is jointly Gaussian. Moreover, any two members belonging to distinct groups are uncorrelated: $\operatorname{Cov}(X^{q}_{j},X^{q'}_{j'})=0$ whenever $q\ne q'$, the covariances being defined and finite by Square-Integrability, Moments, and Covariance Matrix of a Gaussian Random Vector.

When every $J_q$ is a one-element set, this statement recovers Independent Gaussian Random Variables are Jointly Gaussian; the present lemma extends it to arbitrary independent jointly Gaussian groups.