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Quasi-stationary distributions for discrete-state models

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  • van Doorn, Erik A.
  • Pollett, Philip K.

Abstract

This paper contains a survey of results related to quasi-stationary distributions, which arise in the setting of stochastic dynamical systems that eventually evanesce, and which may be useful in describing the long-term behaviour of such systems before evanescence. We are concerned mainly with continuous-time Markov chains over a finite or countably infinite state space, since these processes most often arise in applications, but will make reference to results for other processes where appropriate. Next to giving an historical account of the subject, we review the most important results on the existence and identification of quasi-stationary distributions for general Markov chains, and give special attention to birth–death processes and related models. The question of under what circumstances a quasi-stationary distribution, given its existence, is indeed a good descriptor of the long-term behaviour of a system before evanescence, is addressed as well. We conclude with a discussion of computational aspects, with more details given in a web appendix accompanying this paper.

Suggested Citation

  • 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.
    • Handle: RePEc:eee:ejores:v:230:y:2013:i:1:p:1-14
    DOI: 10.1016/j.ejor.2013.01.032
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    Cited by:

    1. 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.
    2. 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.
    3. 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.
    4. 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.
    5. 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.
    6. Velleret, Aurélien, 2022. "Unique quasi-stationary distribution, with a possibly stabilizing extinction," Stochastic Processes and their Applications, Elsevier, vol. 148(C), pages 98-138.
    7. 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.
    8. Leżaj, Łukasz, 2024. "Non-symmetric stable processes: Dirichlet heat kernel, Martin kernel and Yaglom limit," Stochastic Processes and their Applications, Elsevier, vol. 174(C).
    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.

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