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Quasi-Stationary Asymptotics for Perturbed Semi-Markov Processes in Discrete Time

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  • Mikael Petersson

    (Stockholm University)

Abstract

We consider a discrete time semi-Markov process where the characteristics defining the process depend on a small perturbation parameter. It is assumed that the state space consists of one finite communicating class of states and, in addition, one absorbing state. Our main object of interest is the asymptotic behavior of the joint probabilities of the position of the semi-Markov process and the event of non-absorption as time tends to infinity and the perturbation parameter tends to zero. The main result gives exponential expansions of these probabilities together with a recursive algorithm for computing the coefficients in the expansions. An application to perturbed epidemic SIS models is discussed.

Suggested Citation

  • 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.
    • Handle: RePEc:spr:metcap:v:19:y:2017:i:4:d:10.1007_s11009-016-9530-7
    DOI: 10.1007/s11009-016-9530-7
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    References listed on IDEAS

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    1. 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.
    2. Jung, Brita, 2013. "Exit times for multivariate autoregressive processes," Stochastic Processes and their Applications, Elsevier, vol. 123(8), pages 3052-3063.
    3. , Aisdl, 2007. "Weak convergence of first-rare-event times for semi-Markov processes," OSF Preprints q95cv, Center for Open Science.
    4. Jose Blanchet & Bert Zwart, 2010. "Asymptotic expansions of defective renewal equations with applications to perturbed risk models and processor sharing queues," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 72(2), pages 311-326, October.
    5. Gyllenberg, Mats & S. Silvestrov, Dmitrii, 2000. "Cramer-Lundberg approximation for nonlinearly perturbed risk processes," Insurance: Mathematics and Economics, Elsevier, vol. 26(1), pages 75-90, February.
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