The Donsker's invariance principle (in particular) says that, given the simple random walk in $\mathbb{Z}^d$ $(S_n)_{n \ge 1}$, if we interpolate it linearly by $$ B^{(n)}_t:= \frac{1}{\sqrt{n}} \Big(S_{\lfloor nt \rfloor} + (nt- \lfloor nt \rfloor) S_{\lfloor nt \rfloor+1}\Big), $$ we get $B^{(n)}_\cdot \Longrightarrow B_\cdot$, where $B_\cdot$ is the Brownian motion.
I was wondering if we define $\tau_N=\inf\{t \in \ge 0: |B^{(n)}_t| = 1\}$ and $\tau=\inf\{t \ge 0: |B_t| = 1\}$, do still get that $$ B^{(n)}_{\cdot\wedge \tau_n} \Longrightarrow B_{\cdot \wedge \tau}. $$
Intuitively, I believe it is the case. To prove finite-dimensional convergence I was thinking the following. Given $0=t_0< t_1< \cdots < t_k <t_{k+1}=\infty$, and $f: \mathbb{R}^k \longrightarrow \mathbb{R} $ bounded and continuous, we decompose
\begin{align} \mathbb{E}\Big[f\Big(B^n_{t_1\wedge \tau^n},\dots,B^n_{t_k\wedge \tau^n}\Big)\Big] & = \sum_{j=0}^k \mathbb{E}_N\Big[f\Big(B^n_{t_1\wedge \tau^n},\dots,B^n_{t_k\wedge \tau^n}\Big) \mathbf{1}[t_j <\tau^n \le t_{j+1}]\Big] \\ & = \sum_{j=0}^k \mathbb{E}_N\Big[f\Big(B^n_{t_1},\dots,B^n_{t_j}, B^n_{\tau^n},\dots, B^n_{\tau^n}\Big) \mathbf{1}[t_j <\tau^n \le t_{j+1}]\Big] \\ & = \sum_{j=0}^k \mathbb{E}_N\Big[\tilde f\Big(B^n_{t_1},\dots,B^n_{t_j}, B^n_{\tau^n} \Big) \mathbf{1}[t_j <\tau^n \le t_{j+1}]\Big], \end{align} where $\mathbb E$ is the expectation according to the law of the random walk and $\mathbf{1}$ is the indicator function. I was hoping that I could use that $\tilde f$ is also continuous and bounded and somehow use Donsker's Theorem. However, the dependence in the random time is still a problem.