Consider a sequence of stochastic processes $X^n_t$ on some filtered probability space with measure $\mathbb{P}$. $n\in \mathbb{N}, t\in \mathbb{R}$.
If $X^n_t$ converges (uniformly w.r.t $t$ in compacts) as $n\to \infty$ in $L_2(\mathbb{P})$. Say to $X_t$.
Does this imply that there exists a subsequence $X^{n_{k}}_t$ that converges (uniformly w.r.t $t$ in compacts) $a.s$ to $X_t$?
Thoughts : $L_p(\mathbb{P})$ convergence implies $\mathbb{P}$ convergence, which implies a subsequence converges $a.s$. But I havent seen results concerning uniformity of the convergence.