On the speed of convergence in a boundary problem. I

S. V. Nagaev

Teoriâ veroâtnostej i ee primeneniâ, Tome 15 (1970) no. 2, pp. 179-199

Résumé

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 $$ |W_h-W|<L\frac{c_3^2(K+1)}{\sqrt n}. $$ }