On the speed of convergence in a~boundary problem

A. I. Sakhanenko

Teoriâ veroâtnostej i ee primeneniâ, Tome 19 (1974) no. 2, pp. 416-421

Résumé

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)(t),\quad g_2(0)0(0) \\ |g_i(t+h)-g_i(t)|\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\biggl(\frac kn\biggr),\quad1\le k\le nT\biggr), \\ W(t)=\mathbf P(g_2(t)\xi(t)(t),\quad0), \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].