Mean-Square Limits of Gaussian Random Vectors are Gaussian

Let $(\Omega,\mathcal{F},P)$ be a probability space and let $d\ge1$ be a natural number. For each $k\in\mathbb{N}$ let $(X^{k}_{1},\dots,X^{k}_{d})$ be a Gaussian random vector on $(\Omega,\mathcal{F},P)$, and let $X_1,\dots,X_d$ be square-integrable random variables such that

with $\lVert\cdot\rVert_{2}$ the mean-square norm of Square-Integrable Random Variables and the Mean-Square Inner Product.

Then $(X_1,\dots,X_d)$ is a Gaussian random vector, and its moments are the limits of the moments: with the expectations and covariances (defined and finite for Gaussian random vectors by Square-Integrability, Moments, and Covariance Matrix of a Gaussian Random Vector),

The same conclusions hold if the Gaussian random vectors $(X^{k}_{1},\dots,X^{k}_{d})$ are given only for all sufficiently large $k$ (discard the initial indices and relabel).