Proof of Mean-Square Limits of Affine Combinations Adjoin to a Jointly Gaussian Family

By Jointly Gaussian Families of Random Variables and Gaussian Processes it suffices to show that for every choice of finitely many distinct indices of the combined family, the tuple of corresponding members is a Gaussian random vector. Such a tuple is $(B_{j_1},\dots,B_{j_p},V_{c_1},\dots,V_{c_q})$ with distinct $j$'s in $J$ and distinct $c$'s in $C$ ($p$ or $q$ possibly $0$, not both).

For each $w\in\{1,\dots,q\}$ choose finite affine combinations $S^{w}_n$ of members of $(B_j)$ with $\lVert S^{w}_n-V_{c_w}\rVert_2\to0$, and for the $B_{j_v}$ take the constant sequences $S^{0,v}_n=B_{j_v}$. For fixed $n$, choose finitely many distinct indices of $J$ whose members include $B_{j_1},\dots,B_{j_p}$ and every member of $(B_j)$ occurring in $S^{1}_n,\dots,S^{q}_n$; the corresponding tuple is a Gaussian random vector (Jointly Gaussian Families of Random Variables and Gaussian Processes), and

is its image under an affine map (each coordinate is a real constant plus a finite linear combination of coordinates; for the $B_{j_v}$ the constant is $0$), hence a Gaussian random vector by Affine Transformations of Gaussian Random Vectors are Gaussian. As $n\to\infty$ the coordinates converge in mean square to those of $(B_{j_1},\dots,B_{j_p},V_{c_1},\dots,V_{c_q})$ (constant in the first $p$ coordinates); the limit coordinates are square-integrable (the $B_{j_v}$ by Square-Integrability, Moments, and Covariance Matrix of a Gaussian Random Vector, the $V_{c_w}$ by hypothesis), so Mean-Square Limits of Gaussian Random Vectors are Gaussian shows the limit tuple is a Gaussian random vector. $\blacksquare$