Suppose I have a sequence of stochastic processes $X_{N}(t)$, $N=1,2,3,\ldots$ with mean zero and that I know for every fixed $t$, the random variable $X_{N}(t)$ converges in law to a Gaussian random variable with finite variance as $N \to \infty$. Suppose I also know that the covariance $\mathbb{E}(X_{N}(t_{1})X_{N}(t_{2}))$ converges to $\phi(t_{1},t_{2})$ for every fixed $t_{1}$ and $t_{2}$. Can I now claim that for any natural k, the random vector

$(X_{N}(t_{1}),X_{N}(t_{2}),\ldots,X_{N}(t_{k}))$

converges in law to the k-dimensional Gaussian random vector with covariance matrix $\Sigma_{i,j} = \phi(t_{i},t_{j})$? If not, what are some counterexamples to this statement?