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Approximation of the invariant probability measure of an infinite stochastic matrix

Published online by Cambridge University Press:  01 July 2016

D. Wolf
D. Wolf*
Affiliation:
Technische Universität München
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Abstract

Let P denote an irreducible positive recurrent infinite stochastic matrix with the unique invariant probability measure π. We consider sequences {Pm}m∊N of stochastic matrices converging to P (pointwise), such that every Pm has at least one invariant probability measure πm. The aim of this paper is to find conditions, which assure that at least one of sequences {πm}m∊N converges to π (pointwise). This includes the case where the Pm are finite matrices, which is of special interest. It is shown that there is a sequence of finite stochastic matrices, which can easily be constructed, such that {πm}m∊N converges to π. The conditions given for the general case are closely related to Foster's condition.

Keywords

STOCHASTIC MATRICESINVARIANT PROBABILITY MEASURESAPPROXIMATIONFOSTER'S CONDITIONPOTENTIALS
Type
Research Article
Information
Advances in Applied Probability , Volume 12 , Issue 3 , September 1980 , pp. 710 - 726
Copyright
Copyright © Applied Probability Trust 1980 

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References

Allen, B., Anderson, R. S. and Seneta, E. (1977) Computation of stationary measures for infinite Markov chains. In Studies in the Management Sciences, Vol. 7, ed. Neuts, M. F., North-Holland, Amsterdam.Google Scholar
Chung, K. L. (1960) Markov Chains with Stationary Transition Probabilities. Springer-Verlag, Berlin.CrossRefGoogle Scholar
Cogburn, R. (1975) A uniform theory for sums of Markov chain transition probabilities. Ann. Prob. 3, 191–214.CrossRefGoogle Scholar
Delbrouck, L. E. M. (1971) On stochastic boundedness and stationary measures for Markov processes. J. Math. Anal. Appl. 33, 149–164.CrossRefGoogle Scholar
Foster, F. G. (1953) On the stochastic matrices associated with certain queueing processes. Ann. Math. Statist. 24, 355–360.CrossRefGoogle Scholar
Golub, G. H. and Seneta, E. (1973) Computation of the stationary distribution of an infinite Markov matrix. Bull. Austral. Math. Soc. 8, 333–341.CrossRefGoogle Scholar
Golub, G. H. and Seneta, E. (1974) Computation of the stationary distribution of an infinite stochastic matrix of special form. Bull. Austral. Math. Soc. 10, 255–261.CrossRefGoogle Scholar
Seneta, E. (1967) Finite approximations to infinite non-negative matrices. Proc. Camb. Phil. Soc. 63, 983–992.CrossRefGoogle Scholar
Seneta, E. (1968) Finite approximations to infinite non-negative matrices, II: refinements and applications. Proc. Camb. Phil. Soc. 64, 465–470.CrossRefGoogle Scholar
Seneta, E. (1973) Non-negative Matrices. Allen and Unwin, London.Google Scholar
Tweedie, R. L. (1971) Truncation procedures for non-negative matrices. J. Appl. Prob. 8, 311–320.Google Scholar
Tweedie, R. L. (1976) Criteria for classifying general Markov chains, Adv. Appl. Prob. 8, 737–771.Google Scholar
Wolf, D. (1975) Approximation homogener Markoff-Ketten mit abzählbarem Zustandraum durch solche mit endlichem Zustandsraum. In Proceedings in Operations Research 5, Physica-Verlag, Würzburg.Google Scholar
Wolf, D. (1978) Approximation der invarianten Wahrscheinlichkeitsmasse von Markoffschen Kernen. Dissertation am Fachbereich Mathematik der TH Darmstadt.Google Scholar