Rate of convergence in a boundary problem

Let $\{\xi_i\}_{i\ge 1}$and $\{\tau_i\}_{i\ge 1}$ be the sequences of i. i . d. r. v. 's such that $\tau_1\ge 0$ a. s., $\mathbf E\xi_1=0$, $\mathbf E\xi_1^2=\mathbf E\tau_1=1$ and
$$ S_n=\sum_{i=1}^n\xi_i,\quad T_n=\sum_{i=1}^n\tau_i,\quad\nu(t)=max\{k\ge 0:\,T_k\le t\}. $$
We investigate the rate of convergence
\begin{gather*} \mathbf P\{g^-(n^{-1}T_k)<n^{-1/2}S_k<g^+(n^{-1}T_k),\ k\le \nu(n)\}\to\\ \to\mathbf P\{g^-(t)<w(t)<g^+(t),\ 0\le t\le 1\},\qquad n\to\infty \end{gather*}
where $w(t)$ is a standard Wiener process and $g^\pm(t)$ are Lipschitz functions.