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.

  • $\begingroup$ Guess: NO. Counter-example: Can't think of one immediately. $\endgroup$ – Kavi Rama Murthy Feb 14 at 12:02
  • $\begingroup$ @KaviRamaMurthy Looking through the proofs of $L_p(\mathbb{P})$-convergence implies $\mathbb{P}$-convergence and $\mathbb{P}$-convergence implies existence of $a.s$-convergent sub sequence, the uniformity seems to pass through without trouble? pages 312 and 314 of Probability and Random Processes by R.Grimmett. $\endgroup$ – Monty Feb 14 at 14:45

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