A question regarding Donsker's invariance principle. Donsker states that if $X_1, X_2, ...$ are independent and identically distributed with mean $0$ and variance $\sigma^2$ and if $S_t^n$ is the interpolated and scaled sum of $X_i$ then $(S_t^n)$ converges in $C$ (room of continuous functions) in distribution to the Brownian Motion $B$.
My question concerns a randomly stopped version of $S_t^n$. Let $Y_1, Y_2, ...$ now be i.i.d and nonnegative with mean $\mu$ and variance $\sigma^2$. Let $N_t:=\max{(m \in \mathcal{N} | S_m < t)}$ (with $S_m$ as the unscaled sum of the $Y_i$) and $D_t^n:= \frac{1}{\sigma \sqrt{N_{nt}}} \sum\limits_{i=1}^{N_{nt}}(Y_i-\mu)$ + interpolation term. Does $D_t^n \rightarrow B$ hold in $C$?
Many thanks in advance!