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Subgeometric ergodicity and $\beta$-mixing

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  • Mika Meitz
  • Pentti Saikkonen

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 can not be established. To illustrate our results we derive new subgeometric ergodicity and $\beta$-mixing results for the self-exciting threshold autoregressive model.

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  • Mika Meitz & Pentti Saikkonen, 2019. "Subgeometric ergodicity and $\beta$-mixing," Papers 1904.07103, arXiv.org, revised Apr 2019.
    • Handle: RePEc:arx:papers:1904.07103
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    References listed on IDEAS

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    1. Nummelin, Esa & Tuominen, Pekka, 1983. "The rate of convergence in Orey's theorem for Harris recurrent Markov chains with applications to renewal theory," Stochastic Processes and their Applications, Elsevier, vol. 15(3), pages 295-311, August.
    2. Fort, G. & Moulines, E., 2003. "Polynomial ergodicity of Markov transition kernels," Stochastic Processes and their Applications, Elsevier, vol. 103(1), pages 57-99, January.
    3. Fort, Gersende & Moulines, Eric, 2000. "V-Subgeometric ergodicity for a Hastings-Metropolis algorithm," Statistics & Probability Letters, Elsevier, vol. 49(4), pages 401-410, October.
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