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I ran into the following problem in one of my applied work and would appreciate if someone could kindly shed some lights.

Settings: Let $\{X_n\}$ be a sequence of cadlag processes in the Skorohod space $D[0,1]$ (I could take the stronger assumption that $X_n$ lies in $C[0,1]$, the space of all continuous functions, if needed). We assume that $X_n$ converges in distribution to the Wiener process, $$ X_n \stackrel{d}{\longrightarrow} W $$ Now fix constants $0 \leq l \leq u \leq 1$ and $c > 0$. Define a sequence of stopping times $\tau_n$ by $$ \tau_n = \inf\{ l \leq t \leq u : X_n(t) > c \} \wedge u $$ where $a \wedge b = \min(a, b)$. In other words, $\tau_n$ is the first moment, if exists, when the process $X_n$ moves above a threshold $c$, and equal to $u$ otherwise.

Question:

  1. Does $X_n(\tau_n)$ converge in distribution?

  2. If it converges in distribution, is it possible to derive the closed-form of the cumulative distribution function of the limit?

My guess: I conjecture that $X_n(\tau_n) \stackrel{d}{\longrightarrow} W(\tau)$, where $$ \tau = \inf \{ l \leq t \leq u: W(t) > c \} \wedge u $$ To this end, I have attempted to show $\tau_n$ converges in probability to $\tau$. Were this established, $(X_n, \tau_n)$ would converge in distribution to $(W, \tau)$, and continuous mapping theorem would entail the desired convergence. I couldn't figure it out, however.

Any suggestion will be greatly appreciated!

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    $\begingroup$ I think the Skorohod representation theorem might help; it says you can find another sequence of processes $Y_n$, each having the same distribution as the corresponding $X_n$, with $Y_n$ converging almost surely to a Wiener process. Then it seems you just have to show that if a sequence of paths converges in Skorohod topology, their values at the appropriate stopping times likewise converge. It should be even easier in uniform topology. $\endgroup$ – Nate Eldredge Apr 6 '18 at 20:23
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    $\begingroup$ The limit in distribution seems like it must just be the corresponding stopped Brownian motion, and the distribution of that must be known pretty well. $\endgroup$ – Nate Eldredge Apr 6 '18 at 20:25
  • $\begingroup$ @NateEldredge Thanks much for the suggestion. I have actually considered the use of Skorohod representation theorem to convert the limit problem into an almost sure one. As you suggest, I could take advantage of continuity of the limit and work on the local uniform metric directly. But it appears to me that in general the sequence of stopping times need not converge to the corresponding one of the limit process. $\endgroup$ – Dormire Apr 7 '18 at 9:47
  • $\begingroup$ @NateEldredge I actually imagined that a limit process which attains but not exceeds $c$ on the interval $(l, u)$, and then it seems that we could modify the sequence around the point $\tau$ where the limit achieves $c$ to keep $\liminf \tau_n > \tau$. In this case, it doesn't seem to me that $X_n(\tau_n)$ would converge to $W(\tau)$.. Perhaps the assumptions I impose on $X_n$ are too weak, or perhaps I should utilize some properties stronger than continuity of the Wiener process, which I have had no clue.. $\endgroup$ – Dormire Apr 7 '18 at 9:51
  • $\begingroup$ And yes, you are right: If the limit is Wiener process, it seems that I could make use of the Markov property to determine the c.d.f. $\endgroup$ – Dormire Apr 7 '18 at 9:53

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