Limit of a Gaussian sequence I have a Gaussian stochastic process indexed by positive integers, $(y_t)_{t\geq 1}$. Suppose that 
$$y_t \to z \quad \text{in} \quad L^2$$ 
Then it is true that $z$ is either Gaussian or degenerate. My question is as follows. Assuming that $z$ is Gaussian, is $z,y_1,y_2,\ldots$ a Gaussian sequence, i.e. is every finite vector extracted from this infinite sequence necessarily multivariate Gaussian?

We have to prove that if $a$, $a_1,\dots,a_n$ are real numbers, then 
$Y=az+\sum_{j=1}^ny_j$ is Gaussian. Note that $Y$ is the limit in the $\mathbb L^2$ sense of the sequence $\left(ay_t+\sum_{j=1}^ny_j\right)_{t\geqslant n+1}   $ as $t$ goes to infinity. 
Now, since $\left(y_t\right)_{t\geqslant 1}$ is Gaussian, it follows that for any $t$, the random variable $ay_t+\sum_{j=1}^ny_j$ is Gaussian (or degenerated).   
