Hostname: page-component-cd9895bd7-gbm5v Total loading time: 0 Render date: 2024-12-24T00:44:56.078Z Has data issue: false hasContentIssue false
  • English
  • Français

On the theory of Markov set-chains

Part of: Markov processes

Published online by Cambridge University Press:  01 July 2016

D. J. Hartfiel  and
E. Seneta
D. J. Hartfiel*
Affiliation:
Texas A&M University
E. Seneta*
Affiliation:
University of Sydney
*
* Postal address: Mathematics Department, Texas A&M University, College Station, TX 77843, USA.
** Postal address: School of Mathematics and Statistics, F07, University of Sydney, NSW 2006, Australia.
Get access Rights & Permissions [Opens in a new window]

Abstract

In the theory of homogeneous Markov chains, states are classified according to their connectivity to other states and this classification leads to a classification of the Markov chains themselves. In this paper we classify Markov set-chains analogously, particularly into ergodic, regular, and absorbing Markov set-chains. A weak law of large numbers is developed for regular Markov set-chains. Examples are used to illustrate analysis of behavior of Markov set-chains.

Keywords

WEAK LAW OF LARGE NUMBERSCOEFFICIENT OF ERGODICITYASYMPTOTICS OF MARKOV SET-CHAINS

MSC classification

Primary: 60J10: Markov chains (discrete-time Markov processes on discrete state spaces)
Secondary: 60J20: Applications of Markov chains and discrete-time Markov processes on general state spaces (social mobility, learning theory, industrial processes, etc.)
Type
Research Article
Information
Advances in Applied Probability , Volume 26 , Issue 4 , December 1994 , pp. 947 - 964
Copyright
Copyright © Applied Probability Trust 1994 

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

Berge, C. (1963) Topological Spaces. Oliver and Boyd, Edinburgh.Google Scholar
Bernstein, S. N. (1946) Teoriia Veroiatnostei (Theory of Probabilities) 4th edn, pp. 203–221. Gostehizdat, Moscow.Google Scholar
Hartfiel, D. J. (1981) On the limiting set of stochastic products xA1... Am. Proc. Amer. Math. Soc. 81, 201–206.Google Scholar
Hartfiel, D. J. (1991) Component bounds for Markov set-chain limiting sets. J. Statist. Comput. Simul. 38, 15–24.Google Scholar
Kemeny, J. G. and Snell, J. L. (1960) Finite Markov Chains. Van Nostrand, New York.Google Scholar
Markov, A. A. (1910) Investigation of the general case of trials associated into a chain (in Russian) Zapiski Akad Nauk St Petersburg. In his (1951) lzbrannie Trudy [Selected Works], Akad. Nauk SSSR, Leningrad, pp. 465–507.Google Scholar
Seneta, E. (1981) Nonnegative Matrices and Markov Chains. Springer-Verlag, New York.Google Scholar
Seneta, E. (1984) On the limiting set of nonnegative products. Statist. Prob. Lett. 2, 159–163.Google Scholar