Hostname: page-component-586b7cd67f-gb8f7 Total loading time: 0 Render date: 2024-11-25T10:03:15.537Z Has data issue: false hasContentIssue false
  • English
  • Français

Periodicity in Markov renewal theory

Published online by Cambridge University Press:  01 July 2016

Erhan Çinlar
Erhan Çinlar*
Affiliation:
Northwestern University, Evanston, Illinois
Get access Rights & Permissions [Opens in a new window]

Abstract

In an irreducible Markov renewal process either all states are periodic or none are. In the former case they all have the same period. Periodicity and the period can be determined by direct inspection from the semi-Markov kernel defining the process. The periodicity considerably increases the complexity of the limits in Markov renewal theory especially for transient initial states. Two Markov renewal limit theorems will be given with particular attention to the roles of periodicity and transient states. The results are applied to semi-Markov and semi-regenerative processes.

Keywords

MARKOV RENEWAL THEORYLIMIT THEOREMSPERIODICITYSEMI-REGENERATIVE AND SEMI-MARKOV PROCESSES
Type
Research Article
Information
Advances in Applied Probability , Volume 6 , Issue 1 , March 1974 , pp. 61 - 78
Copyright
Copyright © Applied Probability Trust 1974 

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

[1] Anderson, L. B. (1971) Markov renewal theory on the whole line. Unpublished doctoral dissertation. Northwestern University, Evanston, Illinois.Google Scholar
[2] Çinlar, E. (1969) Markov renewal theory. Adv. Appl. Prob. 1, 123–187.Google Scholar
[3] Çinlar, E. (1969) On semi-Markov processes on arbitrary spaces. Proc. Camb. Phil. Soc. 66, 381–392.Google Scholar
[4] Feller, W. (1966) An Introduction to Probability Theory and its Applications. Vol. II. Wiley, New York.Google Scholar
[5] Neveu, J. (1965) Mathematical Foundations of the Calculus of Probability. Holden-Day, San Francisco.Google Scholar
[6] Pyke, R. and Schaufele, R. (1964) Limit theorems for Markov renewal processes. Ann. Math. Statist. 35, 1746–1764.Google Scholar
[7] Schäl, M. (1970) Markov renewal processes with auxiliary paths. Ann. Math. Statist. 41, 1604–1623.Google Scholar
[8] Smith, W. L. (1955) Regenerative stochastic processes. Proc. Roy. Soc. A 232, 6–31.Google Scholar
[9] Yackel, J. (1966) Limit theorems for semi-Markov processes. Trans. Amer. Math. Soc. 123, 402–424.Google Scholar