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Some sufficient conditions for non-ergodicity of markov chains

Published online by Cambridge University Press:  14 July 2016

Wojciech Szpankowski
Wojciech Szpankowski*
Affiliation:
McGill University
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Abstract

Some sufficient conditions for non-ergodicity are given for a Markov chain with denumerable state space. These conditions generalize Foster's results, in that unbounded Lyapunov functions are considered. Our criteria directly extend the conditions obtained in Kaplan (1979), in the sense that a class of Lyapunov functions is studied. Applications are presented through some examples; in particular, sufficient conditions for non-ergodicity of a multidimensional Markov chain are given.

Keywords

LYAPUNOV FUNCTIONS
Type
Research Papers
Information
Journal of Applied Probability , Volume 22 , Issue 1 , March 1985 , pp. 138 - 147
Copyright
Copyright © Applied Probability Trust 1985 

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References

Chung, K. L. (1967) Markov Chains with Stationary Transition Probabilities. Springer-Verlag, New York.Google Scholar
Foster, F. G. (1953) On stochastic matrices associated with certain queueing processes. Ann. Math. Statist. 24, 355–360.Google Scholar
Kaplan, M. (1979) A sufficient condition for non-ergodicity of a Markov chain. IEEE Trans. Information Theory 25, 470–471.CrossRefGoogle Scholar
Knopp, K. (1956) Infinite Series (in Polish). Warsaw.Google Scholar
Marlin, P. G. (1973) On the ergodic theory of Markov chains. Operat. Res. 21, 617–622.CrossRefGoogle Scholar
Pakes, A. G. (1969) Some conditions for ergodicity and recurrence of Markov chains. Operat. Res. 17, 1058–1061.CrossRefGoogle Scholar
Rosberg, Z. (1981) A note on the ergodicity of Markov chains. J. Appl. Prob. 18, 112–121.CrossRefGoogle Scholar
Sennott, L. I., Humblet, P. and Tweedie, R. L. (1983) Mean drifts and the non-ergodicity of Markov chains. Operat. Res. 21, 783–789.Google Scholar
Szpankowski, W. (1983) Ergodicity aspects of multidimensional Markov chains with application to computer communication systems analysis. In Proc. Internat. Seminar on Modelling and Performance Evaluation Methodology, Paris. Lecture Notes in Control and Information Sciences, Springer-Verlag, Berlin. To appear.Google Scholar
Tweedie, R. L. (1976) Criteria for classifying general Markov chains. Adv. Appl. Prob. 8, 737–771.CrossRefGoogle Scholar