Hostname: page-component-78c5997874-xbtfd Total loading time: 0 Render date: 2024-11-02T23:06:35.747Z Has data issue: false hasContentIssue false

Uniform coupling of non-homogeneous Markov chains

Published online by Cambridge University Press:  14 July 2016

David Griffeath*
Affiliation:
Cornell University

Abstract

The Markov-Dobrushin condition for (weak) ergodicity of non-homogeneous discrete-time Markov chains, and an analogous criterion for continuous chains, are derived by means of coupling techniques.

Type
Research Papers
Copyright
Copyright © Applied Probability Trust 1975 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1] Dobrushin, R. L. (1956) Central limit theorem for nonstationary Markov chains. Theor. Prob. Appl. 1, 6580; 329–383.CrossRefGoogle Scholar
[2] Dobrushin, R. L. (1971) Markov processes with a large number of locally interacting components. Probl. Peredaci Inform. 7, 7087.Google Scholar
[3] Doeblin, W. (1938) Exposé de la théorie des chaînes simples constantes de Markov à un nombre fini d'états. Rev. Math, de l'Union Interbalkanique 2, 77105.Google Scholar
[4] Griffeath, D. (1975) Ergodic theorems for graph interactions. Adv. Appl. Prob. 7, 179194.CrossRefGoogle Scholar
[5] Griffeath, D. (1975) A maximal coupling for Markov chains. Z. Wahrscheinlichkeitsth. 31, 95106.CrossRefGoogle Scholar
[6] Griffeath, D. (1975) Partial coupling and loss of memory for Markov chains. To appear.CrossRefGoogle Scholar
[7] Hajnal, J. (1958) Weak ergodicity in nonhomogeneous Markov chains. Proc. Camb. Phil. Soc. 54, 233246.CrossRefGoogle Scholar
[8] Harris, T. E. (1974) Contact interactions on a lattice. Ann. Prob. 2, 969988.CrossRefGoogle Scholar
[9] Holley, R. and Liggett, T. (1975) Ergodic theorems for weakly interacting particle systems and the voter model. To appear.CrossRefGoogle Scholar
[10] Iosifescu, M. (1966) Conditions nécessaires et suffisantes pour l'ergodicité uniforme des chaînes de Markoff variables et multiples. Rev. Roumaine Math. Pures Appl. 11, 325330.Google Scholar
[11] Iosifescu, M. (1972) On two recent papers on ergodicity in nonhomogeneous Markov chains. Ann. Math. Statist. 43, 17321736.CrossRefGoogle Scholar
[12] Kingman, J. F. C. (1975) Geometrical aspects of the theory of nonhomogeneous Markov chains. Math. Proc. Camb. Phil. Soc. 77, 171183.CrossRefGoogle Scholar
[13] Markov, A. A. (1907) Investigation of an important case of dependent trials. Izv. Akad. Nauk SPB 6, 6180.Google Scholar
[14] Paz, A. (1970) Ergodic theorems for infinite probabilistic tables. Ann. Math. Statist. 41, 539550.CrossRefGoogle Scholar
[15] Seneta, E. (1973) On the historical development of the theory of finite inhomogeneous Markov chains. Proc. Camb. Phil. Soc. 74, 507513.CrossRefGoogle Scholar
[16] Seneta, E. (1973) Non-negative Matrices. Wiley, London.Google Scholar
[17] Vasershtein, L. N. (1969) Markov processes on countable product spaces, describing large systems of automata. Probl. Peredaci Inform. 3, 6472.Google Scholar