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Journal of Statistical Physics

, Volume 177, Issue 4, pp 651–665 | Cite as

Persistence Exponents via Perturbation Theory: AR(1)-Processes

  • Frank Aurzada
  • Marvin KettnerEmail author
Article
  • 26 Downloads

Abstract

For AR(1)-processes \(X_n=\rho X_{n-1}+\xi _n\), \(n\in \mathbb {N}\), where \(\rho \in \mathbb {R}\) and \((\xi _i)_{i\in \mathbb {N}}\) is an i.i.d. sequence of random variables, we study the so-called persistence probabilities \(\mathbb {P}(X_0\ge 0,\ldots , X_N\ge 0)\) for \(N\rightarrow \infty \). For a wide class of Markov processes a recent result (Aurzada et al. in Persistence exponents in Markov chains, arXiv preprint arXiv:1703.06447, 2017) shows that these probabilities decrease exponentially fast and that the rate of decay can be identified as an eigenvalue of some integral operator. We discuss a perturbation technique to determine a series expansion of the eigenvalue in the parameter \(\rho \) for normally distributed AR(1)-processes.

Keywords

Autoregressive process Eigenvalue problem Integral equation Markov chain Persistence Perturbation theory 

Notes

Acknowledgements

We would like to thank two anonymous referees for their comments that helped to improve the exposition of this paper. This work was supported by the Deutsche Forschungsgemeinschaft (Grant AU370/4).

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© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Technische Universität DarmstadtDarmstadtGermany

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