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Quasi-stationary distributions for perturbed discrete time regenerative processes


Author: Mikael Petersson
Original publication: Teoriya Imovirnostei ta Matematichna Statistika, tom 89 (2013).
Journal: Theor. Probability and Math. Statist. 89 (2014), 153-168
MSC (2010): Primary 60K05, 34E10; Secondary 60K25
DOI: https://doi.org/10.1090/S0094-9000-2015-00942-X
Published electronically: January 26, 2015
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Abstract: Non-linearly perturbed discrete time regenerative processes with regenerative stopping times are considered. We define the quasi-stationary distributions for such processes and present conditions for their convergence. Under some additional assumptions, the quasi-stationary distributions can be expanded in asymptotic power series with respect to the perturbation parameter. We give an explicit recurrence algorithm for calculating the coefficients in these asymptotic expansions. Applications to perturbed alternating regenerative processes with absorption and perturbed risk processes are presented.


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  • 1. E. Altman, K. E. Avrachenkov, and R. Núñes-Queija, Perturbation analysis for denumerable Markov chains with application to queueing models, Adv. Appl. Probab. 36 (2004), no. 3, 839-853. MR 2079917 (2005h:60210)
  • 2. J. Darroch and E. Seneta, On quasi-stationary distributions in absorbing discrete-time finite Markov chains, J. Appl. Probab. 2 (1965), 88-100. MR 0179842 (31:4083)
  • 3. J. Darroch and E. Seneta, On quasi-stationary distributions in absorbing continuous-time finite Markov chains, J. Appl. Probab. 4 (1967), 192-196. MR 0212866 (35:3731)
  • 4. E. Ekheden and D. S. Silvestrov, Coupling and explicit rate of convergence in Cramér-Lundberg approximation for reinsurance risk processes, Comm. Statist. Theory Methods 40 (2011), no. 19-20, 3524-3539. MR 2860755
  • 5. E. Englund and D. S. Silvestrov, Mixed large deviation and ergodic theorems for regenerative processes with discrete time, Theory Stoch. Process. 3(19) (1997), no. 1-2, 164-176.
  • 6. W. Feller An Introduction to Probability Theory and Its Applications, Vol. II, Wiley Series in Probability and Statistics, Wiley, New York, 1966, 1971. MR 0270403 (42:5292)
  • 7. J. Grandell, Aspects of Risk Theory, Probability and Its Applications, Springer, New York, 1991. MR 1084370 (92a:62151)
  • 8. M. Gyllenberg and D. S. Silvestrov, Quasi-stationary distributions of a stochastic metapopulation model, J. Math. Biol. 33 (1994), 35-70. MR 1306150 (96b:92012)
  • 9. M. Gyllenberg and D. S. Silvestrov, Quasi-Stationary Phenomena in Nonlinearly Perturbed Stochastic Systems, De Gruyter Expositions in Mathematics, vol. 44, Walter de Gruyter, Berlin, 2008. MR 2456816 (2009k:60005)
  • 10. R. Hassin and M. Haviv, Mean passage times and nearly uncoupled Markov chain, SIAM J. Disc. Math. 5 (1992), 386-397. MR 1172747 (93m:60146)
  • 11. N. V. Kartashov, Asymptotic expansions and inequalities in stability theorems for general Markov chains with relatively bounded perturbations. Stability problems of stochastic models, J. Soviet Math. 40 (1988), no. 4, 509-518. MR 957106 (89j:60089)
  • 12. N. V. Kartashov, Strong Stable Markov Chains, VSP/TViMS, Utrecht/Kiev, 1996. MR 1451375 (99e:60150)
  • 13. R. Z. Khasminskii, G. Yin, and Q. Zhang, Singularly perturbed Markov chains: quasi-stationary distribution and asymptotic expansion, Proceedings of Dynamic Systems and Applications, vol. 2 (Atlanta, GA, 1995), Dynamic, Atlanta, GA, 1996, pp. 301-308. MR 1419542 (97k:34082)
  • 14. J. F. C. Kingman, The exponential decay of Markovian transition probabilities, Proc. London Math. Soc. 13 (1963), 337-358. MR 0152014 (27:1995)
  • 15. V. S. Koroliuk and N. Limnios, Stochastic Systems in Merging Phase Space, World Scientific, Singapore, 2005. MR 2205562 (2007a:60004)
  • 16. G. Latouche, Perturbation analysis of a phase-type queue with weakly correlated arrivals, Adv. Appl. Probab. 20 (1988), 896-912. MR 968004 (89i:60184)
  • 17. M. Petersson and D. S. Silvestrov, Asymptotic expansions for perturbed discrete time renewal equations and regenerative processes, Report 2012:12, Mathematical Statistics, Stockholm University, 2012.
  • 18. E. Seneta and D. Vere-Jones, On quasi-stationary distributions in discrete-time Markov chains with a denumerable infinity of states, J. Appl. Probab. 3 (1966), 403-434. MR 0207047 (34:6863)
  • 19. D. S. Silvestrov, Nonlinearly perturbed Markov chains and large deviations for lifetime functionals, Recent Advances in Reliability Theory: Methodology, Practice and Inference (N. Limnios and M. Nikulin, eds.), Birkhäuser, Boston, 2000, pp. 135-144. MR 1783479
  • 20. D. S. Silvestrov and M. Petersson, Exponential expansions for perturbed discrete time renewal equations, Applied Reliability Engineering and Risk Analysis: Probabilistic Models and Statistical Inference (I. Frenkel, A. Karagrigoriou, A. Kleyner, and A. Lisnianski, eds.), Wiley, Chichester, 2013, pp. 349-362.
  • 21. D. Vere-Jones, Geometric ergodicity in denumerable Markov chains, Quart. J. Math. 13 (1962), 7-28. MR 0141160 (25:4571)
  • 22. G. Yin and D. T. Nguyen, Asymptotic expansions of backward equations for two-time-scale Markov chains in continuous time, Acta Math. Appl. Sin. Engl. Ser. 25 (2009), no. 3, 457-476. MR 2506986 (2010e:60162)
  • 23. G. Yin and Q. Zhang, Discrete-time singularly perturbed Markov chains, Stochastic Modelling and Optimization, Springer, New York, 2003, pp. 1-42. MR 1963518 (2004b:90143)

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Additional Information

Mikael Petersson
Affiliation: Department of Mathematics, Stockholm University, SE-106 91 Stockholm, Sweden
Email: mikpe@math.su.se

DOI: https://doi.org/10.1090/S0094-9000-2015-00942-X
Keywords: Regenerative process, renewal equation, non-linear perturbation, quasi-stationary distribution, asymptotic expansion, risk process
Received by editor(s): November 11, 2012
Published electronically: January 26, 2015
Article copyright: © Copyright 2015 American Mathematical Society

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