---
_id: '33335'
abstract:
- lang: eng
  text: For a class of one-dimensional autoregressive sequences (Xn), we consider
    the tail behaviour of the stopping time T0=min{n≥1:Xn≤0}. We discuss existing
    general analytical approaches to this and related problems and propose a new one,
    which is based on a renewal-type decomposition for the moment generating function
    of T0 and on the analytical Fredholm alternative. Using this method, we show that
    Px(T0=n)∼V(x)Rn0 for some 0<R0<1 and a positive R0-harmonic function V. Further,
    we prove that our conditions on the tail behaviour of the innovations are sharp
    in the sense that fatter tails produce non-exponential decay factors.
author:
- first_name: Günter
  full_name: Hinrichs, Günter
  last_name: Hinrichs
- first_name: Martin
  full_name: Kolb, Martin
  id: '48880'
  last_name: Kolb
- first_name: Vitali
  full_name: Wachtel, Vitali
  last_name: Wachtel
citation:
  ama: Hinrichs G, Kolb M, Wachtel V. Persistence of one-dimensional AR(1)-processes.
    <i>Journal of Theoretical Probability</i>. 2018;33:65–102. doi:<a href="https://link.springer.com/article/10.1007/s10959-018-0850-0">https://link.springer.com/article/10.1007/s10959-018-0850-0</a>
  apa: Hinrichs, G., Kolb, M., &#38; Wachtel, V. (2018). Persistence of one-dimensional
    AR(1)-processes. <i>Journal of Theoretical Probability</i>, <i>33</i>, 65–102.
    <a href="https://link.springer.com/article/10.1007/s10959-018-0850-0">https://link.springer.com/article/10.1007/s10959-018-0850-0</a>
  bibtex: '@article{Hinrichs_Kolb_Wachtel_2018, title={Persistence of one-dimensional
    AR(1)-processes}, volume={33}, DOI={<a href="https://link.springer.com/article/10.1007/s10959-018-0850-0">https://link.springer.com/article/10.1007/s10959-018-0850-0</a>},
    journal={Journal of Theoretical Probability}, publisher={Springer Science + Business
    Media}, author={Hinrichs, Günter and Kolb, Martin and Wachtel, Vitali}, year={2018},
    pages={65–102} }'
  chicago: 'Hinrichs, Günter, Martin Kolb, and Vitali Wachtel. “Persistence of One-Dimensional
    AR(1)-Processes.” <i>Journal of Theoretical Probability</i> 33 (2018): 65–102.
    <a href="https://link.springer.com/article/10.1007/s10959-018-0850-0">https://link.springer.com/article/10.1007/s10959-018-0850-0</a>.'
  ieee: 'G. Hinrichs, M. Kolb, and V. Wachtel, “Persistence of one-dimensional AR(1)-processes,”
    <i>Journal of Theoretical Probability</i>, vol. 33, pp. 65–102, 2018, doi: <a
    href="https://link.springer.com/article/10.1007/s10959-018-0850-0">https://link.springer.com/article/10.1007/s10959-018-0850-0</a>.'
  mla: Hinrichs, Günter, et al. “Persistence of One-Dimensional AR(1)-Processes.”
    <i>Journal of Theoretical Probability</i>, vol. 33, Springer Science + Business
    Media, 2018, pp. 65–102, doi:<a href="https://link.springer.com/article/10.1007/s10959-018-0850-0">https://link.springer.com/article/10.1007/s10959-018-0850-0</a>.
  short: G. Hinrichs, M. Kolb, V. Wachtel, Journal of Theoretical Probability 33 (2018)
    65–102.
date_created: 2022-09-12T07:50:38Z
date_updated: 2022-09-12T07:52:44Z
department:
- _id: '96'
doi: https://link.springer.com/article/10.1007/s10959-018-0850-0
intvolume: '        33'
language:
- iso: eng
page: 65–102
publication: Journal of Theoretical Probability
publication_status: published
publisher: Springer Science + Business Media
status: public
title: Persistence of one-dimensional AR(1)-processes
type: journal_article
user_id: '85821'
volume: 33
year: '2018'
...