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The rate of convergence in Orey's theorem for Harris recurrent Markov chains with applications to renewal theory

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  • Nummelin, Esa
  • Tuominen, Pekka

Abstract

We derive sufficient conditions for [is proportional to] [lambda] (dx)||Pn(x, ·) - [pi]|| to be of order o([psi](n)-1), where Pn (x, A) are the transition probabilities of an aperiodic Harris recurrent Markov chain, [pi] is the invariant probability measure, [lambda] an initial distribution and [psi] belongs to a suitable class of non-decreasing sequences. The basic condition involved is the ergodicity of order [psi], which in a countable state space is equivalent to [Sigma] [psi](n)Pi{[tau]i[greater-or-equal, slanted]n} 0 and [is proportional to] [psi](t)(1- F(t))dt

Suggested Citation

  • 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.
    • Handle: RePEc:eee:spapps:v:15:y:1983:i:3:p:295-311
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    Cited by:

    1. Meitz, Mika & Saikkonen, Pentti, 2022. "Subgeometrically Ergodic Autoregressions," Econometric Theory, Cambridge University Press, vol. 38(5), pages 959-985, October.
    2. Connor, S.B. & Fort, G., 2009. "State-dependent Foster-Lyapunov criteria for subgeometric convergence of Markov chains," Stochastic Processes and their Applications, Elsevier, vol. 119(12), pages 4176-4193, December.
    3. Mika Meitz & Pentti Saikkonen, 2022. "Subgeometrically ergodic autoregressions with autoregressive conditional heteroskedasticity," Papers 2205.11953, arXiv.org, revised Apr 2023.
    4. Mika Meitz & Pentti Saikkonen, 2019. "Subgeometric ergodicity and $\beta$-mixing," Papers 1904.07103, arXiv.org, revised Apr 2019.
    5. Douc, Randal & Fort, Gersende & Guillin, Arnaud, 2009. "Subgeometric rates of convergence of f-ergodic strong Markov processes," Stochastic Processes and their Applications, Elsevier, vol. 119(3), pages 897-923, March.
    6. Iksanov, Alexander & Meiners, Matthias, 2015. "Rate of convergence in the law of large numbers for supercritical general multi-type branching processes," Stochastic Processes and their Applications, Elsevier, vol. 125(2), pages 708-738.
    7. Yong-Hua Mao & Yan-Hong Song, 2022. "Criteria for Geometric and Algebraic Transience for Discrete-Time Markov Chains," Journal of Theoretical Probability, Springer, vol. 35(3), pages 1974-2008, September.
    8. Nummelin, Esa, 1997. "On distributionally regenerative Markov chains," Stochastic Processes and their Applications, Elsevier, vol. 72(2), pages 241-264, December.

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