I am wondering whether I can make this statement: Assume we have centered Gaussian processes $G_n$ and $G$. Furthermore it holds that $Cov(G_n(x),G_n(y))\stackrel{\mathbb{P}}\rightarrow Cov(G(x),G(y))$. As every Gaussian process is determined by its covariance function, is this convergence enough to claim the weak convergence of the processes?
Thanks!