Exit times for some nonlinear autoregressive processes
Pub. online: 25 March 2025
Type: Research Article
Open Access
Open Access
Received
29 August 2024
29 August 2024
Revised
10 January 2025
10 January 2025
Accepted
9 March 2025
9 March 2025
Published
25 March 2025
25 March 2025
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.References
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