On the speed of convergence in a boundary problem. I

Let $\xi_1,\xi_2,\dots$ be a sequence of independent equally distributed random variables with variance 1.
Put $a=\mathbf M\xi_1$, $c_3=\mathbf M|\xi_1-a|^3$.
Let the functions $g_i(t)$, $t\ge0$, $i=1,2$, satisfy the conditions
\begin{gather*} g_2(t)<g_1(t),\quad g_2(0)<0<g_1(0), \\ |g_i(t+h)-g_i(t)|<Kh,\quad h>0, \end{gather*}
where $K$ is some constant.
Put
$$ S_{nk}=\frac1{\sqrt n}\sum_{i=1}^k(\xi_i-a). $$
Let
\begin{gather*} W_n=\mathbf P\{g_2(k/n)<S_{nk}<g_1(k/n),\quad k=\overline{1,n}\}; \\ W=\mathbf P\{g_2(t)<\xi(t)-\xi(0)<g_1(t),\quad0\le t\le1\}, \end{gather*}
where $\xi(t)$ is a process of Brownian motion.
The following assertion is proved.
Theorem.{\em There exists an absolute constant $L$ such that
$$ |W_h-W|<L\frac{c_3^2(K+1)}{\sqrt n}. $$
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