On the exit time of sums of bounded random variables out of a~curve strip

A. A. Novikov

Teoriâ veroâtnostej i ee primeneniâ, Tome 26 (1981) no. 2, pp. 287-301

Résumé

Let $\xi_k$ be independent bounded random variables with $\mathbf E\xi_k=0$, $\mathbf E\xi_k^2=V_k>0$, and $f(k)$, $g(k)$ be some deterministic functions. We investigate the rough asymptotics of the probability $$ \mathbf P\biggl\{\biggl|\sum_1^k\xi_i+f(k)\biggr|\le g(k),\ m\le k\le n\biggr\},\qquad n\to\infty. $$ It is proved that under some assumptions on $f$ and $g$ this asymptotics has the form $$ \operatorname{exp}\biggl\{-\frac{\pi^2}{8}\sum_{k=m}^n V_k g^{-2}(k)(1+o(1))\biggr\} $$ or $$ \operatorname{exp}\biggl\{-1/2\sum_{k=m+1}^nV_k^{-1}|f(k)-f(k-1)|^2(1+o(1))\biggr\}. $$ Our method is based on a change of probability measure which reduces the problem to the case $f(k)\equiv 0$, $g(k)\equiv 1$.