Hostname: page-component-78c5997874-xbtfd Total loading time: 0 Render date: 2024-11-05T03:01:44.214Z Has data issue: false hasContentIssue false
  • English
  • Français

Limit theorems for α-recurrent semi-Markov processes

Published online by Cambridge University Press:  01 July 2016

Esa Nummelin
Esa Nummelin*
Affiliation:
Institute of Mathematics, Helsinki University of Technology
Get access Rights & Permissions [Opens in a new window]

Abstract

In this paper the limit behaviour of α-recurrent Markov renewal processes and semi-Markov processes is studied by using the recent results on the concept of α-recurrence for Markov renewal processes. Section 1 contains the preliminary results, which are needed later in the paper. In Section 2 we consider the limit behaviour of the transition probabilities Pij(t) of an α-recurrent semi-Markov process. Section 4 deals with quasi-stationarity. Our results extend the results of Cheong (1968), (1970) and of Flaspohler and Holmes (1972) to the case in which the functions to be considered are directly Riemann integrable. We also try to correct the errors we have found in these papers. As a special case from our results we consider continuous-time Markov processes in Sections 3 and 5.

Keywords

MARKOV RENEWAL PROCESSESSEMI-MARKOV PROCESSESα-RECURRENCEQUASISTATIONARITYSEMI-MARKOV KERNELMARKOV PROCESSESQ-MATRIX
Type
Research Article
Information
Advances in Applied Probability , Volume 8 , Issue 3 , September 1976 , pp. 531 - 547
Copyright
Copyright © Applied Probability Trust 1976 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Cheong, C. K. (1968) Ergodic and ratio limit theorems for α-recurrent semi-Markov processes. Z. Wahrscheinlichkeitsth. 9, 270–286.Google Scholar
Cheong, C. K. (1970) Quasi-stationary distributions in semi-Markov processes. J. Appl. Prob. 7, 388–399. Correction: J. Appl. Prob. 7, 788.Google Scholar
Chung, K. L. (1967) Markov Chains with Stationary Transition Probabilities, 2nd edn. Springer-Verlag, Berlin.Google Scholar
Çinlar, E. (1969) Markov renewal theory. Adv. Appl. Prob. 1, 123–187.CrossRefGoogle Scholar
Çinlar, E. (1974) Periodicity in Markov renewal theory. Adv. Appl. Prob. 6, 61–78.CrossRefGoogle Scholar
Feller, W. (1971) An Introduction to Probability Theory and its Applications, Vol. 2. Wiley, New York.Google Scholar
Flaspohler, D. C. (1971) Quasi-stationary Distributions. Ph.D. thesis, Rutgers University.Google Scholar
Flaspohler, D. C. and Holmes, P. T. (1972) Additional quasi-stationary distributions for semi-Markov processes. J. Appl. Prob. 9, 671–676.Google Scholar
Kingman, J. F. C. (1963) The exponential decay of Markov transition probabilities. Proc. London Math. Soc. 13, 337–358.Google Scholar
Nummelin, E. (1975) On the concepts of α-recurrence and α-transience for Markov renewal processes. Stoch. Proc. Appl., to appear.Google Scholar
Tweedie, R. L. (1974) Some ergodic properties of the Feller minimal process. Quart. J. Math. 25, 485–495.Google Scholar
Vere-Jones, D. (1962) Geometric ergodicity in denumerable Markov chains. Quart. J. Math. 13, 7–28.Google Scholar
Vere-Jones, D. (1969) Some limit theorems for evanescent processes. Austral. J. Statist. 11, 67–78.CrossRefGoogle Scholar