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On a paper by Doeblin on non-homogeneous Markov chains

Published online by Cambridge University Press:  01 July 2016

Harry Cohn
Harry Cohn*
Affiliation:
University of Melbourne
*
∗Postal address: Department of Statistics, Richard Berry Building, University of Melbourne, Parkville, Victoria 3052, Australia.
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Abstract

In [5] Doeblin considered some classes of finite non-homogeneous Markov chains and gave without proofs several results concerning their asymptotic behaviour. In the present paper we first attempt to make Doeblin's results precise and try to reconstruct his arguments. Subsequently we investigate more general situations, where a state space decomposition is provided by the sets occurring in the representation of the atomic sets of the tail σ-field. We show that Doeblin's notion of an associated chain, as well as considerations regarding the tail σ-field structure of the chain, can be used to solve such cases.

Keywords

FINITE MARKOV CHAINFINAL CLASSCYCLICALLY MOVING SUBCLASSTAIL σ-FIELDATOMIC SETRECURRENCEWHIRLPOOL
Type
Research Article
Information
Advances in Applied Probability , Volume 13 , Issue 2 , June 1981 , pp. 388 - 401
Copyright
Copyright © Applied Probability Trust 1981 

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References

[1] Cohn, H. (1970) On the tail σ-algebra of the finite inhomogeneous Markov chains. Ann. Math. Statist. 41, 2175–2176.Google Scholar
[2] Cohn, H. (1974) A ratio limit theorem for the finite nonhomogeneous Markov chains. Israel J. Math. 19, 329–334.Google Scholar
[3] Cohn, H. (1976) Finite nonhomogeneous Markov chains: Asymptotic behaviour. Adv. Appl. Prob. 8, 502–516.Google Scholar
[4] Cohn, H. (1978) On a paper by Doeblin on nonhomogeneous Markov chains. Research Report Series No. 7. University of Melbourne.Google Scholar
[5] Doeblin, W. (1937) Le cas discontinu de probabilités en chaîne. Publ. Fac. Sci. Univ. Masaryk (Brno) 236, 3–13.Google Scholar
[6] Kemeny, J. G. and Snell, J. L. (1960) Finite Markov Chains. Van Nostrand, Princeton.Google Scholar
[7] Kingman, J. F. C. (1975) Geometrical aspects of the theory of nonhomogeneous Markov chains. Math. Proc. Camb. Phil. Soc. 77, 171–185.CrossRefGoogle Scholar
[8] Kolmogorov, A. N. (1931) Über die analytischen Methoden in der Wahrscheinlichkeitsrechnung. Math. Ann. 104, 415–458.Google Scholar
[9] Loève, M. (1965) Probability Theory, 3rd edn., Van Nostrand, Princeton.Google Scholar
[10] Senchenko, D. V. (1972) The final σ-algebra of an inhomogeneous Markov chain with a finite number of states. Math. Notes. 12, 610–613.Google Scholar
[11] Seneta, E. (1973) On the historical development of the theory of finite nonhomogeneous Markov chains. Proc. Camb. Phil. Soc. 74, 507–513.Google Scholar
[12] Seneta, E. (1973) Non-negative Matrices. Allen and Unwin, London.Google Scholar