Abstract:
Let $\xi_1,\xi_2,\dots$ be a sequence of independent equally distributed random variables with $\mathbf M\xi_1=0$, $\mathbf D\xi_1=1$, $c_3=\mathbf M|\xi_1|^3$. Suppose that 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\text{for all}\quad h>0,
\end{gather*}
where $K$ is some constant.
Put
\begin{gather*}
W_n(t)=\mathbf P\biggl(g_2\biggl(\frac kn\biggr)<\frac1{\sqrt n}\sum_{i=1}^k\xi_i<g_1\biggl(\frac kn\biggr),\quad1\le k\le nT\biggr),
\\
W(t)=\mathbf P(g_2(t)<\xi(t)<g_1(t),\quad0<t<T),
\end{gather*}
where $\xi(t)$ is a Brownian motion process, $\xi(0)=0$.
The following assertions are proved.
Theorem 1.Theore exists an absolute constant $L_1$ such that $$
|W_n(1)-W(1)|\le L_1\frac{(K+1)c_3}{\sqrt n}.
$$
Theorem 2.There exists an absolute constant $L_2 \le L_1$ such that $$
|W_n(\infty)-W(\infty)|\le L_2\frac{Kc_3}{\sqrt n}.
$$
Theorem 1 is a generalization of the main result of [1] and [2].
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