Modern Stochastics: Theory and Applications

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Göran Högnäs Brita Jung

Pub. online: 25 Mar 2025 Type: Research Article

Open Access

Journal: Modern Stochastics: Theory and Applications

Abstract

The expected exit time from the interval

$[-1,1]$ is investigated for an autoregressive process defined recursively by

${X_{n+1}^{\varepsilon }}=f\big({X_{n}^{\varepsilon }}\big)+\varepsilon {\xi _{n+1}},\hspace{1em}n=0,1,2,\dots ,\hspace{2.5pt}{X_{0}}=0.$

Here, ε is a small positive parameter,

$f:\mathbb{R}\mapsto \mathbb{R}$ is usually a contractive function and

${\{{\xi _{n}}\}_{n\ge 1}}$ is a sequence of i.i.d. random variables. In this paper, previous results for a linear function

$f(x)=ax$ are extended to more general cases, with the main focus on piecewise linear functions.

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