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:
Does $X_n(\tau_n)$ converge in distribution?
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!