if I have a series of gaussian processes :
($W_{t}^{n}$ is gaussian process for every n) and I know that for every t there exist $W_t $ s.t $ E|W_t^n-W_t|^2\to0 $as $n\to \infty$. how can I show that $W_t$ is also gaussian process?
hint I got : prove that $Ee^{iu\cdot \bar{W_t^n}}\to E e^{iu\cdot \bar{W_t}} $as $ n\to \infty $ $$u\in R^k ,\bar{W_t}=(W_{t_1},..W_{t_k}),\ \cdot \text{ is inner product} $$ thank you
The characteristic function of $(W_{t_1}^n,\dots,W_{t_d}^n)$ can be computed by the following date: the mean of this vector, and the covariance matrix.
The convergence of $W_t^n\to W_t$ in $L^2$ shows that $\mathbb E[W_{t_i}^n]\to \mathbb E[W_{t_i}]$ and that $$\lim_{n\to \infty}\operatorname{Cov}\left(W_{t_i}^n,W_{t_j}^n \right)=\operatorname{Cov}\left(W_{t_i},W_{t_j} \right) ,\quad i,j\in \{1,\dots,d\} . $$