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A note on simultaneous recurrence conditions on a set of denumerable stochastic matrices

Published online by Cambridge University Press:  14 July 2016

A. Federgruen ,
A. Hordijk  and
H. C. Tijms
A. Federgruen
Affiliation:
Mathematisch Centrum, Amsterdam
A. Hordijk
Affiliation:
Rijksuniversiteit Leiden
H. C. Tijms
Affiliation:
Vrije Universiteit, Amsterdam
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Abstract

In this paper we consider a set of denumerable stochastic matrices where the parameter set is a compact metric space. We give a number of simultaneous recurrence conditions on the stochastic matrices and establish equivalences between these conditions. The results obtained generalize corresponding results in Markov chain theory to a considerable extent and have applications in stochastic control problems.

Keywords

COMPACT METRIC SET OF DENUMBERABLE STOCHASTIC MATRICESSIMULTANEOUS RECURRENCE CONDITIONSDOEBLIN CONDITIONSCRAMBLING CONDITIONQUASI-COMPACTNESS CONDITIONEQUIVALENCES
Type
Short Communications
Information
Journal of Applied Probability , Volume 15 , Issue 4 , December 1978 , pp. 842 - 847
Copyright
Copyright © Applied Probability Trust 1978 

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References

[1] Doob, J. L. (1953) Stochastic Processes. Wiley, New York.Google Scholar
[2] Federgruen, A. and Tijms, H. C. (1978) The optimality equation in average cost denumerable state semi-Markov decision problems, recurrency conditions and algorithms. J. Appl. Prob. 15, 356–373.CrossRefGoogle Scholar
[3] Federgruen, A., Hordijk, A. and Tijms, H. C. (1978) Recurrence conditions in denumer able state Markov decision processes In Dynamic Programming and its Applications , ed. Puterman, M. L. Academic Press, New York.Google Scholar
[4] Hordijk, A. (1974) Dynamic Programming and Markov Potential Theory. Mathematical Centre Tract No. 51, Mathematisch Centrum, Amsterdam.Google Scholar
[5] Huang, C., Isaacson, D. and Vinograde, B. (1976) The rate of convergence of certain nonhomogeneous Markov chains. Z. Wahrscheinlichkeitsth. 35, 141–146.CrossRefGoogle Scholar
[6] Isaacson, D. and Madsen, R. (1974) Positive columns for stochastic matrices. J. Appl. Prob. 11, 829–835.CrossRefGoogle Scholar
[7] Neveu, J. (1965) Mathematical Foundations of the Calculus of Probability. Holden-Day, San Francisco.Google Scholar
[8] Rudin, W. (1964) Principles of Mathematical Analysis , 2nd edn. McGraw-Hill, New York.Google Scholar