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August 2021 Persistence exponents in Markov chains
Frank Aurzada, Sumit Mukherjee, Ofer Zeitouni
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Ann. Inst. H. Poincaré Probab. Statist. 57(3): 1411-1441 (August 2021). DOI: 10.1214/20-AIHP1114

Abstract

We prove the existence of the persistence exponent

logλ:=limn 1 nlog Pμ(X0S,,XnS)

for a class of time homogeneous Markov chains {Xi}i0 taking values in a Polish space, where S is a Borel measurable set and μ is an initial distribution. Focusing on the case of AR(p) and MA(q) processes with p,qN and continuous innovation distribution, we study the existence of λ and its continuity in the parameters of the AR and MA processes, respectively, for S=R0. For AR processes with log-concave innovation distribution, we prove the strict monotonicity of λ. Finally, we compute new explicit exponents in several concrete examples.

Nous démontrons l’existence de l’exposant de persistance

logλ:=limn 1 nlog Pμ(X0S,,XnS)

pour une classe de chaines de Markov {Xi}i0 homogènes en temps avec valeurs dans un espace Polonais, où S est un ensemble Borélien et μ est une distribution initiale. En nous concentrant sur le cas de processus de type AR(p) ou MA(q) avec p,qN et une distribution d’innovation continue, nous étudions l’existence de l’exposant λ et sa continuité par rapport au paramètres des processus AR et MA, pour S=R0. Pour des processus AR ayant une distribution d’innovations qui est log-concave, nous démontrons la monotonicité stricte de λ. Finalement, nous calculons explicitement les exposants dans quelques exemples concrets.

Acknowledgements

We thank the American Institute of Mathematics (AIM) for hosting SQuaREs which brought the authors together and at which occasions the initial questions for this project were discussed. We thank Ohad Feldheim for help with the proof of Proposition 2.3, and Amir Dembo, Naomi Feldheim, and Fuchang Gao for their helpful comments and suggestions. We would also like to thank the AE and an anonymous referee, whose comments have greatly improved the presentation of this paper.

Citation

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Frank Aurzada. Sumit Mukherjee. Ofer Zeitouni. "Persistence exponents in Markov chains." Ann. Inst. H. Poincaré Probab. Statist. 57 (3) 1411 - 1441, August 2021. https://doi.org/10.1214/20-AIHP1114

Information

Received: 19 October 2018; Revised: 6 April 2020; Accepted: 14 October 2020; Published: August 2021
First available in Project Euclid: 22 July 2021

Digital Object Identifier: 10.1214/20-AIHP1114

Subjects:
Primary: 60F10 , 60J05
Secondary: 45C05‎ , 47A75

Keywords: ARMA , eigenvalue problem , integral equation , large deviations , Markov chain , Persistence , quasi-stationary distribution

Rights: Copyright © 2021 Association des Publications de l’Institut Henri Poincaré

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Vol.57 • No. 3 • August 2021
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