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Harmonic functions for a class of Markov chains

Part of: Markov processes

Published online by Cambridge University Press:  09 April 2009

H. Cohn
H. Cohn
Affiliation:
Department of Statistics University of MelbourneParkville, Australia
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Abstract

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A class of Markov chains is considered for which a certain property of the tail events makes bounded harmonic functions obtainable from bounded space-time harmonic functions. Applications to almost surely convergent Markov chains are given and, in particular, a representation of Martin-Doob-Hunt type is derived for all bounded harmonic functions of a finite mean supercritical branching process.

MSC classification

Secondary: 60J05: Discrete-time Markov processes on general state spaces
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 28 , Issue 4 , December 1979 , pp. 413 - 422
Copyright
Copyright © Australian Mathematical Society 1979

References

Blackwell, D. (1955), ‘On transient Markov processes with a countable number of states and stationary transition probabilities’, Ann. Math. Statist. 26, 654–658.CrossRefGoogle Scholar
Brown, B. M. (1977), ‘The tail σ-field of a branching process’ (unpublished).Google Scholar
Cohn, H. (1977a), ‘On the norming constants occurring in convergent Markov chains’, Bull. Austral. Math. Soc. 17, 193–205.CrossRefGoogle Scholar
Cohn, H. (1977b), ‘Almost sure convergence of branching processes’, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 38, 73–81.CrossRefGoogle Scholar
Cohn, H. (1979), ‘On the invariant events of a Markov chain’, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 48, 81–96.CrossRefGoogle Scholar
Dubuc, S. (1970), ‘La fonction de Greene d'un processus de Galton-Watson’, Studia Math. 34, 69–87.CrossRefGoogle Scholar
Dubuc, S. (1971), ‘Problèmes relatifs à l'iteration des fonctions suggérés par les processus en cascade’, Ann. Inst. Fourier, Grenoble, 21, 171–251.CrossRefGoogle Scholar
Heyde, C. C. (1970), ‘Extension of a result by Seneta for the supercritical branching process’, Ann. Math. Statist. 41, 739–742.CrossRefGoogle Scholar
Lootgieter, J. C. (1977), ‘La σ-algèbre asymptotique d'une chaine de Galton-Watson’, Ann. Inst. Henri Poincari, 13 (3), 193–230.Google Scholar
Pakes, A. G. (1978), ‘On the age distribution of a Markov chain’, J. Appl. Probab. 15, 65–77.CrossRefGoogle Scholar
Riesz, F. and Sz.-Nagy, B. (1955), Functional analysis (F. Ungar, New York).Google Scholar
Revuz, D. (1975), Markov chains (North-Holland, Amsterdam).Google Scholar
Seneta, E. (1968), ‘On recent theorems oncerning the supercritical Galton-Watson process’, Ann. Math. Statist. 39, 2098–2102.CrossRefGoogle Scholar
Schuh, H.-J. and Barbour, A. (1977), ‘On the asymptotic behaviour of branching processes with infinite mean’, Advances in Appl. Probability 9, 681–723.CrossRefGoogle Scholar