Hostname: page-component-cd9895bd7-q99xh Total loading time: 0 Render date: 2024-12-27T08:16:56.806Z Has data issue: false hasContentIssue false
  • English
  • Français

Information flow in discrete Markov systems

Published online by Cambridge University Press:  14 July 2016

D. A. Dawson
D. A. Dawson*
Affiliation:
Carleton University, Ottawa
Get access Rights & Permissions [Opens in a new window]

Abstract

Infinite systems of interacting Markov chains are investigated. Some basic concepts of the ergodic theory of such systems are first presented. In particular, the question of the uniqueness and multiplicity of invariant probability measures is considered. In the case of one-dimensional systems, the question is studied in detail by investigating the flow of information throughout the system and some criteria for the uniqueness of invariant probability measures are obtained.

Keywords

INTERACTING MARKOV CHAINSINFORMATION FLOWERGODIC CLASSESINVARIANT PROBABILITY MEASURES
Type
Research Papers
Information
Journal of Applied Probability , Volume 10 , Issue 1 , March 1973 , pp. 63 - 83
Copyright
Copyright © Applied Probability Trust 1973 

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] Averintzev, M. B. (1970) On a method of describing discrete parameter random fields. Problemy Peredci Informacii 6, 100–109.Google Scholar
[2] Abramson, N. (1963) Information Theory and Coding. McGraw-Hill, New York.Google Scholar
[3] Beboutoff, M. (1942) Markov chains with a compact state space. Rec. Math. (Mat. Sb. N. S.) 10(52), 213–238.Google Scholar
[4] Belyaev, Yu. K., Gromak, Yu. I. and Malyshev, V. A. (1969) Invariant random Boolean fields. Mathematicheskie Zametki 6, 555–566.Google Scholar
[5] Cowan, J. D. (1968) Statistical mechanics of nervous nets. Neural Networks, ed. Cainiello, E. R. Springer-Verlag, Berlin.Google Scholar
[6] Glauber, R. J. (1963) Time-dependent statistics of the Ising model. J . Math. and Phys. 4, 294–307.CrossRefGoogle Scholar
[7] Hammersley, J. M. and Handscomb, D. C. (1964) Monte Carlo Methods. Methuen, London.Google Scholar
[8] Iosifescu, M. and Theodorescu, R. (1968) Random Processes and Learning. Springer-Verlag, New York-Heidelberg-Berlin.Google Scholar
[9] Jacobs, K. (1963) Ergodic theory. Lecture notes, Aarhus University.Google Scholar
[10] Leigh, E. C. (1968) The ecological role of Volterra's equations. Mathematical Problems in Biology, ed. Gerstenhaber, M. Amer. Math. Soc., Providence, Rhode Island.Google Scholar
[11] Rosenblatt, M. (1967) Transition probability operators. Fifth Berkeley Symposium, Vol 2, Part 2, 473–483. University of California, Berkeley.Google Scholar
[12] Ruelle, D. (1969) Statistical Mechanics. W. A. Benjamin, New York-Amsterdam.Google Scholar
[13] Spitzer, F. (1971) Markov random fields and Gibbs ensembles. Amer. Math. Monthly 78, 142–155.Google Scholar
[14] Spitzer, F. (1970) Interaction Markov processes. Advances in Math. 5, 246–290.Google Scholar
[15] Toom, A. L. (1968) A family of uniform nets of formal neurons. Sov, Math. 9, 1338–1341,Google Scholar