|
Teoriya Veroyatnostei i ee Primeneniya, 1981, Volume 26, Issue 2, Pages 287–301
(Mi tvp2511)
|
|
|
|
This article is cited in 3 scientific papers (total in 3 papers)
On the exit time of sums of bounded random variables out of a curve strip
A. A. Novikov Moscow
Abstract:
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
$f(k)\equiv 0$, $g(k)\equiv 1$.
Received: 29.04.1980
Citation:
A. A. Novikov, “On the exit time of sums of bounded random variables out of a curve strip”, Teor. Veroyatnost. i Primenen., 26:2 (1981), 287–301; Theory Probab. Appl., 26:2 (1982), 279–292
Linking options:
https://www.mathnet.ru/eng/tvp2511 https://www.mathnet.ru/eng/tvp/v26/i2/p287
|
|