Let $S_T$ be a cadlag process on $[l,u]$, where $0 < l < u < 1$, such that $$ S_T \stackrel{d}{\longrightarrow} S $$ where $S(t) \equiv \frac{W(t)}{\sqrt{t}}$ and $W$ is a Wiener process. It follows that at any $t$, the asymptotic limit of $S_T(t)$ is standard normal.
Now I am wondering if the asymptotic limit of $S_T(\tau_T)$ is still standard normal, where $\tau_T$ is a random variable taking values in $(l,u)$ and independent of the processes. I guess the answer is yes, and here is my attempt: Let $F_T$ be the cumulative distribution function of $\tau_T$. By independence, $$ \mathbb{E}[\exp(i \theta S_T(\tau_T))] = \int_l^u \mathbb{E}[\exp(i \theta S_T(t))] dF_T(t) $$ The integrand converges pointwise to $\exp(-\frac{1}{2} \theta^2)$. Thus as long as there is a suitable convergence theorem we can conclude the proof.
My question is then: is there such a suitable convergence theorem that I can use? Or maybe something is wrong with my reasoning? I would greatly appreciate, if there is any error, that you point out what additional assumptions are required to establish the desired conclusion.
Many thanks!