Timeline for Convergence of the probability that hitting times being infinity
Current License: CC BY-SA 4.0
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| Oct 27, 2020 at 23:44 | comment | added | Nate Eldredge | Maybe an even worse example is just to let $X_n = 1/n$ independent of both $t$ and $\omega$, with $X=0$. Then $X_n \to X$ uniformly and almost surely, which is about the strongest mode one could imagine, but $\mathbb{P}(\tau_n = \infty)=1$ and $\mathbb{P}(\tau = \infty)=0$. So I don't think you will get your desired result without knowing something more specific about the processes themselves. | |
| Oct 27, 2020 at 23:24 | history | asked | user128095 | CC BY-SA 4.0 |