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A sequence of jointly measurable stochastic processes $\{X_n\}_{n \in \mathbb N}$ converges to the limit $X$ uniformly on compacts in probability (ucp) if

$$P\left(\sup_{s\le t}\vert X_{n}(s)-X(s)\vert>\epsilon\right)\rightarrow 0 $$

as ${n\rightarrow\infty}$ for each ${t,\epsilon>0}$.

Is it true that if $X_n \rightarrow X$ ucp, then there exists a sub-sequence of $X_n$ that converges uniformly on compacts almost surely to $X$ ?

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Yes. Let $Y_{nk}=\sup_{s\leq k} |X_n (s)-X(s)|$. Then $Y_{nk} \to 0$ in probability as $ n \to \infty$ for each $k$. For each $k$ there is a subsequence that converges almost surely. By a diagonal procedure we can find one subsequence $\{n_i\}$ of the integers such that $Y_{{n_i} k} \to 0$ almost surely for each $k$. This implies that $X_{n_i} \to X$ ucp.

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  • $\begingroup$ I think you meant that $X_{n_i}$ converges uniformly on compacts almost surely to $X$ at the end. $\endgroup$
    – W. Volante
    May 7, 2018 at 18:40
  • $\begingroup$ Yes, that is what I meant. $\endgroup$ May 8, 2018 at 23:52

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