IDEAS home Printed from https://ideas.repec.org/a/eee/spapps/v148y2022icp98-138.html
   My bibliography  Save this article

Unique quasi-stationary distribution, with a possibly stabilizing extinction

Author

Listed:
  • Velleret, Aurélien

Abstract

We establish sufficient conditions for exponential convergence to a unique quasi-stationary distribution in the total variation norm. These conditions also ensure the existence and exponential ergodicity of the Q-process, the process conditioned upon never being absorbed. The technique relies on a coupling procedure that is related to Harris recurrence (for Markov Chains). It applies to general continuous-time and continuous-space Markov processes. The main novelty is that we modulate each coupling step depending both on a final horizon of time (for survival) and on the initial distribution. By this way, we could notably include in the convergence a dependency on the initial condition. As an illustration, we consider a continuous-time birth–death process with catastrophes and a diffusion process describing a (localized) population adapting to its environment.

Suggested Citation

  • Velleret, Aurélien, 2022. "Unique quasi-stationary distribution, with a possibly stabilizing extinction," Stochastic Processes and their Applications, Elsevier, vol. 148(C), pages 98-138.
  • Handle: RePEc:eee:spapps:v:148:y:2022:i:c:p:98-138
    DOI: 10.1016/j.spa.2022.02.004
    as

    Download full text from publisher

    File URL: http://www.sciencedirect.com/science/article/pii/S0304414922000400
    Download Restriction: Full text for ScienceDirect subscribers only

    File URL: https://libkey.io/10.1016/j.spa.2022.02.004?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. Champagnat, Nicolas & Villemonais, Denis, 2021. "Lyapunov criteria for uniform convergence of conditional distributions of absorbed Markov processes," Stochastic Processes and their Applications, Elsevier, vol. 135(C), pages 51-74.
    2. van Doorn, Erik A. & Pollett, Philip K., 2013. "Quasi-stationary distributions for discrete-state models," European Journal of Operational Research, Elsevier, vol. 230(1), pages 1-14.
    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. Economou, A. & Gómez-Corral, A. & López-García, M., 2015. "A stochastic SIS epidemic model with heterogeneous contacts," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 421(C), pages 78-97.
    2. Mikael Petersson, 2017. "Quasi-Stationary Asymptotics for Perturbed Semi-Markov Processes in Discrete Time," Methodology and Computing in Applied Probability, Springer, vol. 19(4), pages 1047-1074, December.
    3. Michel Benaïm & Nicolas Champagnat & William Oçafrain & Denis Villemonais, 2023. "Transcritical Bifurcation for the Conditional Distribution of a Diffusion Process," Journal of Theoretical Probability, Springer, vol. 36(3), pages 1555-1571, September.
    4. Popov, Serguei & Shcherbakov, Vadim & Volkov, Stanislav, 2022. "Linear competition processes and generalized Pólya urns with removals," Stochastic Processes and their Applications, Elsevier, vol. 144(C), pages 125-152.
    5. Artalejo, J.R. & Economou, A. & Lopez-Herrero, M.J., 2015. "The stochastic SEIR model before extinction: Computational approaches," Applied Mathematics and Computation, Elsevier, vol. 265(C), pages 1026-1043.
    6. Corujo, Josué, 2021. "Dynamics of a Fleming–Viot type particle system on the cycle graph," Stochastic Processes and their Applications, Elsevier, vol. 136(C), pages 57-91.
    7. Claude Lefèvre & Matthieu Simon, 2022. "On the Risk of Ruin in a SIS Type Epidemic," Methodology and Computing in Applied Probability, Springer, vol. 24(2), pages 939-961, June.
    8. He, Guoman & Zhang, Hanjun & Zhu, Yixia, 2019. "On the quasi-ergodic distribution of absorbing Markov processes," Statistics & Probability Letters, Elsevier, vol. 149(C), pages 116-123.
    9. Ravner, Liron, 2014. "Equilibrium arrival times to a queue with order penalties," European Journal of Operational Research, Elsevier, vol. 239(2), pages 456-468.

    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:148:y:2022:i:c:p:98-138. 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: 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.