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Two classification theorems of states of Markov chains

Published online by Cambridge University Press:  14 July 2016

C. W. Kim
C. W. Kim*
Affiliation:
Simon Fraser University
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Summary

Two theorems on Markov chains, both of which already appear in the literature: the classification of the states into the set of all non-recurrent (transient) states and recurrent classes, and the corresponding classification for idempotent Markov chains due to Doob, are proved from the viewpoint of ergodic theory.

Type
Short Communications
Information
Journal of Applied Probability , Volume 7 , Issue 2 , August 1970 , pp. 490 - 496
Copyright
Copyright © Applied Probability Trust 1970 

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References

[1] Chacon, R. V. (1962) Identification of the limit of operator averages. J. Math. Mech. 11, 961–968.Google Scholar
[2] Chung, K. L. (1967) Markov Chains with Stationary Transition Probabilities. 2nd ed. Springer-Verlag, New York.Google Scholar
[3] Doob, J. L. (1942) Topics in the theory of Markoff chains. Trans. Amer. Math. Soc. 52, 37–64.CrossRefGoogle Scholar
[4] Feldman, J. (1962) Subinvariant measures for Markoff operators. Duke Math. J. 29, 71–98.Google Scholar
[5] Feller, W. (1968) An Introduction to Probability Theory and its Applications. Vol. 1. 3rd ed. Wiley, New York.Google Scholar
[6] Foguel, S. R. (1969) The Ergodic Theory of Markov Processes. Van Nostrand Reinhold, Cincinnati.Google Scholar
[7] Hopf, E. (1954) The general temporally discrete Markoff process. J. Rat. Mech. Anal. 3, 13–45.Google Scholar
[8] Kim, C. W. (1968) A generalization of Ito's theorem concerning the pointwise ergodic theorem. Ann. Math. Statist. 39, 2145–2148.CrossRefGoogle Scholar
[9] Neveu, J. (1965) Mathematical Foundations of the Calculus of Probability. Holden-Day, San Francisco.Google Scholar