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Subgeometric ergodicity and β-mixing

Part of: Markov processes Ergodic theory

Published online by Cambridge University Press:  16 September 2021

Mika Meitz Open the ORCID record for Mika Meitz [Opens in a new window]  and
Pentti Saikkonen
Pentti Saikkonen*
Affiliation:
University of Helsinki
*
**Postal address: Department of Mathematics and Statistics, University of Helsinki, PO Box 68, FI–00014 University of Helsinki, Finland. Email address: pentti.saikkonen@helsinki.fi
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Abstract

It is well known that stationary geometrically ergodic Markov chains are $\beta$ -mixing (absolutely regular) with geometrically decaying mixing coefficients. Furthermore, for initial distributions other than the stationary one, geometric ergodicity implies $\beta$ -mixing under suitable moment assumptions. In this note we show that similar results hold also for subgeometrically ergodic Markov chains. In particular, for both stationary and other initial distributions, subgeometric ergodicity implies $\beta$ -mixing with subgeometrically decaying mixing coefficients. Although this result is simple, it should prove very useful in obtaining rates of mixing in situations where geometric ergodicity cannot be established. To illustrate our results we derive new subgeometric ergodicity and $\beta$ -mixing results for the self-exciting threshold autoregressive model.

Keywords

Markov chainrate of convergencemixing coefficientsubgeometric ratesubexponential ratepolynomial rateSETAR model

MSC classification

Primary: 60J05: Discrete-time Markov processes on general state spaces
Secondary: 37A25: Ergodicity, mixing, rates of mixing
Type
Original Article
Information
Journal of Applied Probability , Volume 58 , Issue 3 , September 2021 , pp. 594 - 608
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Copyright
© The Author(s), 2021. Published by Cambridge University Press on behalf of Applied Probability Trust

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Footnotes

The supplementary material for this article can be found at http://doi.org/10.1017/jpr.2020.108.

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