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Exponential ergodicity in derived Markov chains

Published online by Cambridge University Press:  14 July 2016

Jozef L. Teugels
Jozef L. Teugels*
Affiliation:
University of Louvain, Belgium
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Extract

A general proposition is proved stating that the exponential ergodicity of a stationary Markov chain is preserved for derived Markov chains as defined by Cohen [2], [3]. An application to a certain type of continuous time Markov chains is included.

Type
Research Papers
Information
Journal of Applied Probability , Volume 5 , Issue 3 , December 1968 , pp. 669 - 678
Copyright
Copyright © Applied Probability Trust 1968 

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References

[1] Chung, K. L. (1960) Markov Chains with Stationary Transition Probabilities. Springer Verlag.Google Scholar
[2] Cohen, J. W. (1962) On derived and nonstationary Markov chains. Theor. Prob. Appl. 7, 402–423.Google Scholar
[3] Cohen, J. W. (1962) Derived Markov chains. I, II, III. Nederl. Akad. Wetensch. Proc. Ser. A. 65, 55–92.Google Scholar
[4] Feller, W. (1966) An Introduction to Probability Theory and its Applications. II. Wiley, New York.Google Scholar
[5] Hille, E. and Phillips, R. S. (1957) Functional analysis and semigroups. Amer. Math. Soc. Coll. Publ. 31.Google Scholar
[6] Karlin, S. (1966) A First Course in Stochastic Processes. Academic Press, New York.Google Scholar
[7] Karlin, S. and Mcgregor, J. L. (1958) Many server queueing processes with Poisson input and exponential service times. Pacific J. Math. 8, 87–118.Google Scholar
[8] Kendall, D. G. (1959) Unitary dilations of Markov transition operators and the corresponding integral representation for transition probability matrices. Probability and Statistics. Editor Grenander, U. Almqvist & Wiksell, Stockholm.Google Scholar
[9] Kendall, D. G. (1960) Geometric ergodicity and the theory of queues. Mathematical Methods in the Social Sciences. Editors Arrow, K., Karlin, S., Suppes, P. Stanford University Press, California.Google Scholar
[10] Kingman, J. F. C. (1963) The exponential decay of Markov transition probabilities. Proc. Lond. Math. Soc. 13, 337–358.CrossRefGoogle Scholar
[11] Kingman, J. F. C. (1963) Ergodic properties of continuous-time Markov processes and their discrete skeletons. Proc. Lond. Math. Soc. 13, 593–604.Google Scholar
[12] Miller, H. D. (1966) Geometric ergodicity in a class of denumerable Markov chains. Z. Wahrscheinlichkeitsth. 4, 354–373.Google Scholar
[13] Teugels, J. L. (1967) On the rate of convergence in renewal and Markov renewal processes. Mimeo Series 107, Department of Statistics, Purdue University.Google Scholar
[14] Vere-Jones, D. (1962) Geometric ergodicity in denumerable Markov chains. Quart. J. Math. 13, 7–28.Google Scholar