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

Let $(\Omega,\mathcal{F},P)$ be a probability space, let $(B_j)_{j\in J}$ be a jointly Gaussian family of random variables, and let $(V_c)_{c\in C}$ be a family of square-integrable random variables such that every $V_c$ is a mean-square limit of finite affine combinations of members of $(B_j)_{j\in J}$ — that is, of random variables of the form $a_0+\sum_{q=1}^{p}a_qB_{j_q}$ with real coefficients.

Then the combined family consisting of all $B_j$ ($j\in J$) and all $V_c$ ($c\in C$), indexed by the disjoint union of $J$ and $C$, is jointly Gaussian.