Hostname: page-component-cd9895bd7-mkpzs Total loading time: 0 Render date: 2024-12-23T05:07:12.407Z Has data issue: false hasContentIssue false
  • English
  • Français

Criteria for ergodicity, exponential ergodicity and strong ergodicity of Markov processes

Published online by Cambridge University Press:  14 July 2016

R. L. Tweedie
R. L. Tweedie*
Affiliation:
CSIRO Division of Mathematics and Statistics, Melbourne
*
∗Postal address: CSIRO Division of Mathematics and Statistics, P.O. Box 310, South Melbourne, Victoria 3205, Australia.
Get access Rights & Permissions [Opens in a new window]

Abstract

For regular Markov processes on a countable space, we provide criteria for the forms of ergodicity in the title in terms of the existence of solutions to inequalities involving the Q-matrix of the process. An application to birth-death processes is given.

Keywords

Q-MATRIX INEQUALITIESJUMP-CHAINGEOMETRIC ERGODICITYHITTING TIMESBIRTH-DEATH PROCESSES
Type
Research Papers
Information
Journal of Applied Probability , Volume 18 , Issue 1 , March 1981 , pp. 122 - 130
Copyright
Copyright © Applied Probability Trust 

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

Chung, K. L. (1967) Markov Chains with Stationary Transition Probabilities, 2nd edn, Springer-Verlag, Berlin.Google Scholar
Çinlar, E. (1975) Introduction to Stochastic Processes. Prentice-Hall, New Jersey.Google Scholar
Isaacson, D. and Arnold, B. (1978) Strong ergodicity for continuous-time Markov chains. J. Appl. Prob. 15, 699–706.CrossRefGoogle Scholar
Isaacson, D. and Tweedie, R. L. (1978) Criteria for strong ergodicity of Markov chains. J. Appl. Prob. 15, 87–95.Google Scholar
Kingman, J. F. C. (1963) Ergodic properties of continuous-time Markov processes and their discrete skeletons. Proc. London Math. Soc. (3) 13, 593–604.Google Scholar
Kingman, J. F. C. (1964) The stochastic theory of regenerative events. Z. Wahrscheinlichkeitsth. 2, 180–224.Google Scholar
Mertens, J. F., Samuel-Cahn, E. and Zamit, S. (1978) Necessary and sufficient conditions for recurrence and transience of Markov chains, in terms of inequalities. J. Appl. Prob. 15, 848–851.CrossRefGoogle Scholar
Pakes, A. G. (1969) Some conditions for ergodicity and recurrence of Markov chains. Operat. Res. 17, 1058–1061.Google Scholar
Popov, N. (1977) Conditions for geometric ergodicity of countable Markov chains. Soviet Math. Dokl. 18, 676–679.Google Scholar
Reuter, G. E. H. (1957) Denumerable Markov processes and the associated contraction semigroups on l. Acta Math. 97, 1–46.Google Scholar
Reuter, G. E. H. (1961) Competition processes. Proc. 4th Berkeley Symp. Math. Statist. Prob. 2, 421–430.Google Scholar
Tuominen, P. and Tweedie, R. L. (1979) Exponential ergodicity in Markovian queueing and dam models. J. Appl. Prob. 16, 867–880.Google Scholar
Tweedie, R. L. (1975) Sufficient conditions for regularity, recurrence and ergodicity of Markov processes. Math. Proc. Camb. Phil. Soc. 78, 125–136.Google Scholar