Let $\mathcal{G}$ be the class of all stochastic processes that are Markovian and Gaussian at the same time. Let $Y$ be a process in the closure of $\mathcal{G}$, with the sense of convergence given in $L^2$. Is $Y$ necessarily Gauss-Markov? Is there any characterization of the set $\mathcal{G}$ in the stochastic process literature?

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