Asymptotics for Quasi-stationary Distributions of Perturbed Discrete Time Semi-Markov Processes

  • Mikael PeterssonEmail author
Conference paper
Part of the Springer Proceedings in Mathematics & Statistics book series (PROMS, volume 179)


In this paper we study quasi-stationary distributions of non-linearly perturbed semi-Markov processes in discrete time. This type of distributions are of interest for analysis of stochastic systems which have finite lifetimes but are expected to persist for a long time. We obtain asymptotic power series expansions for quasi-stationary distributions and it is shown how the coefficients in these expansions can be computed from a recursive algorithm. As an illustration of this algorithm, we present a numerical example for a discrete time Markov chain.


Semi-Markov process Perturbation Quasi-stationary distribution Asymptotic expansion Renewal equation Markov chain 


  1. 1.
    Altman, E., Avrachenkov, K.E., Núñez-Queija, R.: Perturbation analysis for denumerable Markov chains with application to queueing models. Adv. Appl. Prob. 36, 839–853 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Avrachenkov, K.E., Haviv, M.: The first Laurent series coefficients for singularly perturbed stochastic matrices. Linear Algebra Appl. 386, 243–259 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Avrachenkov, K.E., Filar, J.A., Howlett, P.G.: Analytic Perturbation Theory and Its Applications. SIAM, Philadelphia (2013)CrossRefzbMATHGoogle Scholar
  4. 4.
    Cheong, C.K.: Ergodic and ratio limit theorems for \(\alpha \)-recurrent semi-Markov processes. Z. Wahrscheinlichkeitstheorie verw. Geb. 9, 270–286 (1968)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Cheong, C.K.: Quasi-stationary distributions in semi-Markov processes. J. Appl. Prob. 7, 388–399 (1970). (Correction in J. Appl. Prob. 7, 788)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Courtois, P.J., Louchard, G.: Approximation of eigencharacteristics in nearly-completely decomposable stochastic systems. Stoch. Process. Appl. 4, 283–296 (1976)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Darroch, J.N., Seneta, E.: On quasi-stationary distributions in absorbing discrete-time finite Markov chains. J. Appl. Prob. 2, 88–100 (1965)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Delebecque, F.: A reduction process for perturbed Markov chains. SIAM J. Appl. Math. 43, 325–350 (1983)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Englund, E., Silvestrov, D.S.: Mixed large deviation and ergodic theorems for regenerative processes with discrete time. In: Jagers, P., Kulldorff, G., Portenko, N., Silvestrov, D. (eds.) Proceedings of the Second Scandinavian-Ukrainian Conference in Mathematical Statistics, vol. I, Umeå (1997) (Also in: Theory Stoch. Process. 3(19), no. 1–2, 164–176 (1997))Google Scholar
  10. 10.
    Flaspohler, D.C., Holmes, P.T.: Additional quasi-stationary distributions for semi-Markov processes. J. Appl. Prob. 9, 671–676 (1972)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Gaĭtsgori, V.G., Pervozvanskiĭ, A.A.: Aggregation of states in a Markov chain with weak interaction. Cybernetics 11, 441–450 (1975)CrossRefGoogle Scholar
  12. 12.
    Gyllenberg, M., Silvestrov, D.S.: Quasi-stationary distributions of stochastic metapopulation model. J. Math. Biol. 33, 35–70 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Gyllenberg, M., Silvestrov, D.S.: Quasi-stationary phenomena for semi-Markov processes. In: Janssen, J., Limnios, N. (eds.) Semi-Markov Models and Applications, pp. 33–60. Kluwer, Dordrecht (1999)CrossRefGoogle Scholar
  14. 14.
    Gyllenberg, M., Silvestrov, D.S.: Quasi-Stationary Phenomena in Nonlinearly Perturbed Stochastic Systems. De Gruyter Expositions in Mathematics, vol. 44. Walter de Gruyter, Berlin (2008)CrossRefzbMATHGoogle Scholar
  15. 15.
    Hassin, R., Haviv, M.: Mean passage times and nearly uncoupled Markov chains. SIAM J. Discrete Math. 5(3), 386–397 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Kingman, J.F.C.: The exponential decay of Markov transition probabilities. Proc. Lond. Math. Soc. 13, 337–358 (1963)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Latouche, G.: First passage times in nearly decomposable Markov chains. In: Stewart, W.J. (ed.) Numerical Solution of Markov Chains. Probability: Pure and Applied, vol. 8, pp. 401–411. Marcel Dekker, New York (1991)Google Scholar
  18. 18.
    Latouche, G., Louchard, G.: Return times in nearly-completely decomposable stochastic processes. J. Appl. Prob. 15, 251–267 (1978)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Petersson, M.: Asymptotics of ruin probabilities for perturbed discrete time risk processes. In: Silvestrov, D., Martin-Löf, A. (eds.) Modern Problems in Insurance Mathematics. EAA Series, pp. 95–112. Springer, Cham (2014)Google Scholar
  20. 20.
    Petersson, M.: Quasi-stationary distributions for perturbed discrete time regenerative processes. Theory Probab. Math. Statist. 89, 153–168 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Petersson, M.: Quasi-stationary asymptotics for perturbed semi-Markov processes in discrete time. Research Report 2015:2, Department of Mathematics, Stockholm University, 36 pp. (2015)Google Scholar
  22. 22.
    Petersson, M.: Asymptotic expansions for moment functionals of perturbed discrete time semi-Markov processes. In: Silvestrov, S., Rančić, M. (eds.) Engineering Mathematics II. Algebraic, Stochastic and Analysis Structures for Networks, Data Classification and Optimization. Springer, Berlin (2016)Google Scholar
  23. 23.
    Schweitzer, P.J.: Perturbation theory and finite Markov chains. J. Appl. Prob. 5, 401–413 (1968)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Seneta, E., Vere-Jones, D.: On quasi-stationary distributions in discrete-time Markov chains with a denumerable infinity of states. J. Appl. Prob. 3, 403–434 (1966)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Silvestrov, D.S., Petersson, M.: Exponential expansions for perturbed discrete time renewal equations. In: Frenkel, I., Karagrigoriou, A., Lisnianski, A., Kleyner, A. (eds.) Applied Reliability Engineering and Risk Analysis: Probabilistic Models and Statistical Inference, pp. 349–362. Wiley, Chichester (2013)CrossRefGoogle Scholar
  26. 26.
    Silvestrov, D., Silvestrov S.: Asymptotic expansions for stationary distributions of perturbed semi-Markov processes. Research Report 2015:9, Department of Mathematics, Stockholm University, 75 pp. (2015)Google Scholar
  27. 27.
    Simon, H.A., Ando, A.: Aggregation of variables in dynamic systems. Econometrica 29, 111–138 (1961)CrossRefzbMATHGoogle Scholar
  28. 28.
    Stewart, G.W.: On the sensitivity of nearly uncoupled Markov chains. In: Stewart, W.J. (ed.) Numerical Solution of Markov Chains. Probability: Pure and Applied, vol. 8, pp. 105–119. Marcel Dekker, New York (1991)Google Scholar
  29. 29.
    van Doorn, E.A., Pollett, P.K.: Quasi-stationary distributions for discrete-state models. Eur. J. Oper. Res. 230, 1–14 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Vere-Jones, D.: Geometric ergodicity in denumerable Markov chains. Q. J. Math. 13, 7–28 (1962)MathSciNetCrossRefzbMATHGoogle Scholar
  31. 31.
    Yin, G., Zhang, Q.: Continuous-Time Markov Chains and Applications. A Singular Perturbation Approach. Applications of Mathematics, vol. 37. Springer, New York (1998)Google Scholar
  32. 32.
    Yin, G., Zhang, Q.: Discrete-time singularly perturbed Markov chains. In: Yao, D.D., Zhang, H., Zhou, X.Y. (eds.) Stochastic Modelling and Optimization, pp. 1–42. Springer, New York (2003)CrossRefGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2016

Authors and Affiliations

  1. 1.Department of MathematicsStockholm UniversityStockholmSweden

Personalised recommendations