IDEAS home Printed from https://ideas.repec.org/a/spr/jotpro/v25y2012i1d10.1007_s10959-011-0384-1.html
   My bibliography  Save this article

Asymptotics of Markov Additive Chains on a Half-Plane: A Ratio Limit Theorem

Author

Listed:
  • Aziz Khanchi

    (University of Ottawa)

Abstract

Consider a Markov additive chain (V,Z) with a negative horizontal drift on a half-plane. We provide the limiting distribution of Z when V passes a threshold for the first time, as V tends to infinity. Our contribution is to allow the Markovian part of an associated twisted Markov chain to be null recurrent or transient. The positive recurrent case was treated by Kesten [Ann. Probab. 2 (1974), 355–386]. Moreover, a ratio limit will be established for a transition kernel with unbounded jumps.

Suggested Citation

  • Aziz Khanchi, 2012. "Asymptotics of Markov Additive Chains on a Half-Plane: A Ratio Limit Theorem," Journal of Theoretical Probability, Springer, vol. 25(1), pages 62-76, March.
    • Handle: RePEc:spr:jotpro:v:25:y:2012:i:1:d:10.1007_s10959-011-0384-1
    DOI: 10.1007/s10959-011-0384-1
    as

    Download full text from publisher

    File URL: http://link.springer.com/10.1007/s10959-011-0384-1
    File Function: Abstract
    Download Restriction: Access to the full text of the articles in this series is restricted.
    File URL: https://libkey.io/10.1007/s10959-011-0384-1?utm_source=ideas
    LibKey link: if access is restricted and if your library uses this service, LibKey will redirect you to where you can use your library subscription to access this item
    ---><---

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

    References listed on IDEAS

    as
    1. Högnäs, Göran, 1997. "On the quasi-stationary distribution of a stochastic Ricker model," Stochastic Processes and their Applications, Elsevier, vol. 70(2), pages 243-263, October.
    2. Masakiyo Miyazawa, 2009. "Tail Decay Rates in Double QBD Processes and Related Reflected Random Walks," Mathematics of Operations Research, INFORMS, vol. 34(3), pages 547-575, August.
    Full references (including those not matched with items on IDEAS)

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Toshihisa Ozawa & Masahiro Kobayashi, 2018. "Exact asymptotic formulae of the stationary distribution of a discrete-time two-dimensional QBD process," Queueing Systems: Theory and Applications, Springer, vol. 90(3), pages 351-403, December.
    2. Yanting Chen & Richard J. Boucherie & Jasper Goseling, 2020. "Necessary conditions for the compensation approach for a random walk in the quarter-plane," Queueing Systems: Theory and Applications, Springer, vol. 94(3), pages 257-277, April.
    3. Toshihisa Ozawa, 2022. "Tail asymptotics in any direction of the stationary distribution in a two-dimensional discrete-time QBD process," Queueing Systems: Theory and Applications, Springer, vol. 102(1), pages 227-267, October.
    4. Wendi Li & Yuanyuan Liu & Yiqiang Q. Zhao, 2019. "Exact tail asymptotics for fluid models driven by an M/M/c queue," Queueing Systems: Theory and Applications, Springer, vol. 91(3), pages 319-346, April.
    5. Yanting Chen & Richard J. Boucherie & Jasper Goseling, 2016. "Invariant measures and error bounds for random walks in the quarter-plane based on sums of geometric terms," Queueing Systems: Theory and Applications, Springer, vol. 84(1), pages 21-48, October.
    6. Ramanan, Kavita & Zeitouni, Ofer, 1999. "The quasi-stationary distribution for small random perturbations of certain one-dimensional maps," Stochastic Processes and their Applications, Elsevier, vol. 84(1), pages 25-51, November.
    7. Toshihisa Ozawa, 2021. "Asymptotic properties of the occupation measure in a multidimensional skip-free Markov-modulated random walk," Queueing Systems: Theory and Applications, Springer, vol. 97(1), pages 125-161, February.
    8. Kamil Demirberk Ünlü & Ali Devin Sezer, 2020. "Excessive backlog probabilities of two parallel queues," Annals of Operations Research, Springer, vol. 293(1), pages 141-174, October.
    9. Khanchi, Aziz & Lamothe, Gilles, 2011. "Simulating tail asymptotics of a Markov chain," Statistics & Probability Letters, Elsevier, vol. 81(9), pages 1392-1397, September.
    10. Yiqiang Q. Zhao, 2022. "The kernel method tail asymptotics analytic approach for stationary probabilities of two-dimensional queueing systems," Queueing Systems: Theory and Applications, Springer, vol. 100(1), pages 95-131, 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:spr:jotpro:v:25:y:2012:i:1:d:10.1007_s10959-011-0384-1. 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.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with 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: Sonal Shukla or Springer Nature Abstracting and Indexing (email available below). General contact details of provider: http://www.springer.com .

    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.