IDEAS home Printed from https://ideas.repec.org/a/eee/spapps/v50y1994i1p37-56.html
   My bibliography  Save this article

On the Markov renewal theorem

Author

Listed:
  • Alsmeyer, Gerold

Abstract

Let (S, £) be a measurable space with countably generated [sigma]-field £ and (Mn, Xn)n[greater-or-equal, slanted]0 a Markov chain with state space S x and transition kernel :S x ( [circle times operator] )-->[0, 1]. Then (Mn,Sn)n[greater-or-equal, slanted]0, where Sn = X0+...+Xn for n[greater-or-equal, slanted]0, is called the associated Markov random walk. Markov renewal theory deals with the asymptotic behavior of suitable functionals of (Mn,Sn)n[greater-or-equal, slanted]0 like the Markov renewal measure [Sigma]n[greater-or-equal, slanted]0P((Mn,Sn)[epsilon]Ax (t+B)) as t-->[infinity] where A[epsilon] and B denotes a Borel subset of . It is shown that the Markov renewal theorem as well as a related ergodic theorem for semi-Markov processes hold true if only Harris recurrence of (Mn)n[greater-or-equal, slanted]0 is assumed. This was proved by purely analytical methods by Shurenkov [15] in the one-sided case where (x,Sx[0,[infinity])) = 1 for all x[epsilon]S. Our proof uses probabilistic arguments, notably the construction of regeneration epochs for (Mn)n[greater-or-equal, slanted]0 such that (Mn,Xn)n[greater-or-equal, slanted]0 is at least nearly regenerative and an extension of Blackwell's renewal theorem to certain random walks with stationary, 1-dependent increments.

Suggested Citation

  • Alsmeyer, Gerold, 1994. "On the Markov renewal theorem," Stochastic Processes and their Applications, Elsevier, vol. 50(1), pages 37-56, March.
  • Handle: RePEc:eee:spapps:v:50:y:1994:i:1:p:37-56
    as

    Download full text from publisher

    File URL: http://www.sciencedirect.com/science/article/pii/0304-4149(94)90146-5
    Download Restriction: Full text for ScienceDirect subscribers only
    ---><---

    As the access to this document is restricted, you may want to search for a different version of it.

    Citations

    Citations are extracted by the CitEc Project, subscribe to its RSS feed for this item.
    as


    Cited by:

    1. Masakiyo Miyazawa, 2011. "Light tail asymptotics in multidimensional reflecting processes for queueing networks," TOP: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 19(2), pages 233-299, December.
    2. Fuh, Cheng-Der, 2021. "Asymptotic behavior for Markovian iterated function systems," Stochastic Processes and their Applications, Elsevier, vol. 138(C), pages 186-211.
    3. Hansen, Niels Richard & Jensen, Anders Tolver, 2005. "The extremal behaviour over regenerative cycles for Markov additive processes with heavy tails," Stochastic Processes and their Applications, Elsevier, vol. 115(4), pages 579-591, April.
    4. Basrak, Bojan & Conroy, Michael & Olvera-Cravioto, Mariana & Palmowski, Zbigniew, 2022. "Importance sampling for maxima on trees," Stochastic Processes and their Applications, Elsevier, vol. 148(C), pages 139-179.
    5. Gobet, Emmanuel & Menozzi, Stéphane, 2010. "Stopped diffusion processes: Boundary corrections and overshoot," Stochastic Processes and their Applications, Elsevier, vol. 120(2), pages 130-162, February.
    6. Haas, Bénédicte & Stephenson, Robin, 2018. "Bivariate Markov chains converging to Lamperti transform Markov additive processes," Stochastic Processes and their Applications, Elsevier, vol. 128(10), pages 3558-3605.
    7. Gobet, Emmanuel & Menozzi, Stéphane, 2004. "Exact approximation rate of killed hypoelliptic diffusions using the discrete Euler scheme," Stochastic Processes and their Applications, Elsevier, vol. 112(2), pages 201-223, August.
    8. Dmitry Korshunov, 2008. "The Key Renewal Theorem for a Transient Markov Chain," Journal of Theoretical Probability, Springer, vol. 21(1), pages 234-245, March.
    9. Alsmeyer, Gerold & Hoefs, Volker, 2002. "Markov renewal theory for stationary (m+1)-block factors: convergence rate results," Stochastic Processes and their Applications, Elsevier, vol. 98(1), pages 77-112, March.
    10. Fuh, Cheng-Der & Zhang, Cun-Hui, 2000. "Poisson equation, moment inequalities and quick convergence for Markov random walks," Stochastic Processes and their Applications, Elsevier, vol. 87(1), pages 53-67, May.
    11. Alsmeyer, Gerold, 1996. "Superposed continuous renewal processes A Markov renewal approach," Stochastic Processes and their Applications, Elsevier, vol. 61(2), pages 311-322, February.

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:eee:spapps:v:50:y:1994:i:1:p:37-56. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    We have no bibliographic references for this item. You can help adding them by using this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: Catherine Liu (email available below). General contact details of provider: http://www.elsevier.com/wps/find/journaldescription.cws_home/505572/description#description .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.