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Weak ergodicity in non-homogeneous Markov chains

Published online by Cambridge University Press:  24 October 2008

J. Hajnal  and
M. S. Bartlett
J. Hajnal
Affiliation:
London School of Economics and Political Science
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Abstract

The paper deals with weak ergodicity, i.e. the tendency for a chain to ‘forget’ the distant past. This may occur in non-homogeneous chains even if the probabilities of being in a given state do not tend to a limit as the number of trials increases.

Investigation of the general conditions of weak ergodicity leads to the definition of a special subclass of regular matrices. Conditions for chains involving regular matrices are also given (§§2,3).

Section 4 discusses similarities in asymptotic behaviour between chains whose transition matrices differ only by small amounts.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 54 , Issue 2 , April 1958 , pp. 233 - 246
Copyright
Copyright © Cambridge Philosophical Society 1958

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References

REFERENCES

(1)Fréchet, M. Théorie des Événements en Chaîne dans le Cas d'un Nombre Fini d'Etats Possibles, in Traité du Calcul des Probabilités et de sea Applications (edited by Borel, E.), vol. I (Paris, 1938).Google Scholar
(2)Hajnal, J.The ergodic properties of non-homogeneous finite Markov chains. Proc. Camb. Phil. Soc. 52 (1956), 67.CrossRefGoogle Scholar
(3)Sarymsakov, T. A.On the ergodic principle for non-stationary Markov chains. Dokl. Akad. Nauk. SSSR (N.S.) 90 (1953), 25.Google Scholar