Nonlinearly perturbed regenerative processes and pseudo-stationary phenomena for stochastic systems

Stochastic Processes and their Applications

Volumn 86, Issue 1, 2000, Pages 1-27

  Gyllenberg, Mats ^a   Silvestrov, Dmitrii S ^a,b,c  

^a UNIVERSITY OF TURKU   (Finland)

^b UMEÅ UNIVERSITY   (Sweden)

^c MÄLARDALEN UNIVERSITY   (Sweden)

Author keywords

60J10; 60J15; 60J27; 60K15; Ergodic theorem; Highly reliable queuing system; Large deviation theorem; Metapopulation; Renewal equation; Semi Markov process

Indexed keywords

EID: 0042415565     PISSN: 03044149     EISSN: None     Source Type: Journal    
DOI: 10.1016/S0304-4149(99)00084-8     Document Type: Article

References (30)

1. Darroch J., Seneta E. On quasi-stationary distributions in absorbing discrete-time finite Markov chains. J. Appl. Probab. 2:1965;88-100.
2. Day J.R., Possingham H.P. A stochastic metapopulation model with variability in patch size and position. Theor. Popul. Biol. 48:1995;333-360.
3. Diekmann, O., Heesterbeek, J.A.P., 1999. Mathematical Epidemiology of Infectious Diseases: Model Building, Analysis and Interpretation. Wiley, New York.
4. Feller, W., 1966. An Introduction to Probability Theory and its Applications, Vol. II. Wiley, New York.
5. Gyllenberg M., Hanski I. Habitat deterioration, habitat destruction and metapopulation persistence in a heterogeneous landscape. Theor. Pop. Biol. 52:1997;198-215.
6. Gyllenberg, M., Hanski, I., Hastings, A., 1997. Structured metapopulation models. In: Hanski, I., Gilpin M. (Eds.), Metapopulation Dynamics: Ecology, Genetics and Evolution. Academic Press, London, pp. 93-122.
7. Gyllenberg M., Silvestrov D.S. Quasi-stationary distributions of a stochastic metapopulation model. J. Math. Biol. 33:1994;35-70.
8. Gyllenberg, M., Silvestrov, D.S., 1997. Exponential asymptotics for perturbed renewal equations and pseudo-stationary phenomena for stochastic systems. Research Report 3, Dept. Math. Statist., Umeå Univ., ISSN 1401-730X.
9. Gyllenberg, M., Silvestrov, D.S., 1999. Quasi-stationary phenomena for semi-Markov processes. In: Janssen, J., Limnios, N. (Eds.), The Advances on Semi-Markov Models: Theory and Applications. Kluwer, Dordrecht, pp. 33-60.
10. Hanski, I., 1999. Metapopulation Ecology. Oxford University Press, Oxford.
11. Hanski, I., Gilpin, M.E., 1997. Metapopulation Biology: Ecology, Genetics, and Evolution. Academic Press, London.
12. Hassin R., Haviv M. Mean passage times and nearly uncoupled Markov chains. SIAM J. Discrete Math. 5:1992;386-397.
13. Hethcote, H.W., Yorke, J.A., 1984. Gonorrhea Transmission Dynamics and Control. Springer, Berlin.
14. Kartashov, N.V., 1996. Strong Stable Markov Chains. VSP, Utrecht and TBiMC Publishers, Kiev.
15. Keilson J. A limit theorem for passage times in ergodic regenerative processes. Ann. Math. Statist. 37:1966;866-870.
16. Keilson, J., 1978. Markov Chain Models - Rarity and Exponentiality. Springer, New York.
17. Kingman J.F. The exponential decay of Markovian transition probabilities. Proc. London Math. Soc. 13:1963;337-358.
18. Korolyuk V.S. Asymptotical behaviour of sojourn time of semi-Markov processes in a subset of states. Ukraine Math. J. 21:1969;842-845.
19. Korolyuk, V.S., Turbin, A.F., 1976. Semi-Markov Processes and its Applications. Naukova Dumka, Kiev.
20. Kovalenko, I.N., Kuznetzov, N., Pegg, P.A., 1997. Mathematical Theory of Reliability of Time Dependent Systems with Practical Applications. Wiley, New York.
21. Schweitzer P.J. Perturbation theory and finite Markov chains. J. Appl. Probab. 5:1968;401-413.
22. Seneta E., Vere-Jones D. On quasi-stationary distributions in discrete-time Markov chains with a denumerable infinity of states. J. Appl. Probab. 3:1966;403-434.
23. Silvestrov D.S. On one generalization of the renewal theorem. Dokl. Akad. Nauk USSR Ser. A. 11:1976;978-982.
24. Silvestrov, D.S., 1978. The renewal theorem in a series scheme. Theory Probab. Math. Statist. Part 1: 18, 144-161, Part 2: 20, 97-116.
25. Silvestrov, D.S., 1979. The renewal theorem in a series scheme. Theory Probab. Math. Statist. Part 1: 18, 144-161, Part 2: 20, 97-116.
26. Silvestrov D.S. Exponential asymptotic for perturbed renewal equations. Theory Probab. Math. Statist. 52:1995;143-153.
27. Simon H.A., Ando A. Aggregation of variables in dynamic systems. Econometrica. 29:1961;11-138.
28. Solov'ev A.D. Asymptotical behaviour of the first occurrence time of a rare event in a regenerative process. Izv. Akad. Nauk SSSR, Tekhn. Kibern. 6:1971;79-89.
29. Vere-Jones D. Geometric ergodicity in denumerable Markov chains. Quart. J. Math. 13:1962;7-28.
30. Yaglom A.M. Certain limit theorems of the theory of branching processes. Dokl. Acad. Nauk SSSR. 56:1947;795-798.