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The robustness of positive recurrence and recurrence of Markov chains under perturbations of the transition probabilities

Published online by Cambridge University Press:  14 July 2016

Richard L. Tweedie
Richard L. Tweedie*
Affiliation:
Australian National Universitycor1corresp
*
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Abstract

In many Markov chain models, the immediate characteristic of importance is the positive recurrence of the chain. In this note we investigate whether positivity, and also recurrence, are robust properties of Markov chains when the transition laws are perturbed. The chains we consider are on a fairly general state space : when specialised to a countable space, our results are essentially that, if the transition matrices of two irreducible chains coincide on all but a finite number of columns, then positivity of one implies positivity of both; whilst if they coincide on all but a finite number of rows and columns, recurrence of one implies recurrence of both. Examples are given to show that these results (and their general analogues) cannot in general be strengthened.

Keywords

ERGODIC MARKOV CHAINSRECURRENT MARKOV CHAINSPERTURBATIONS
Type
Research Papers
Information
Journal of Applied Probability , Volume 12 , Issue 4 , December 1975 , pp. 744 - 752
Copyright
Copyright © Applied Probability Trust 1975 

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References

[1] Lamperti, J. (1960) Criteria for the recurrence or transience of stochastic process I. J. Math. Anal. Appl. 1, 314–330.CrossRefGoogle Scholar
[2] Lamperti, J. (1963) Criteria for stochastic processes II: passage-time moments. J. Math. Anal. Appl. 7, 127–145.CrossRefGoogle Scholar
[3] Miller, H. D. (1966) Geometric ergodicity in a class of denumerable Markov chains. Z. Wahrscheinlichkeltsth. 4, 354–373.CrossRefGoogle Scholar
[4] Orey, S. (1971) Lecture Notes on Limit Theorems for Markov Chain Transition Probabilities. Van Nostrand Reinhold, London.Google Scholar
[5] Pollard, D. B. and Tweedie, R. L. (1975a) R-theory for Markov chains on a topological state space I. J. Lond. Math. Soc. To appear.Google Scholar
[6] Pollard, D. B. and Tweedie, R. L. (1975b) R-theory for Markov chains on a topological state space II. Submitted for publication .Google Scholar
[7] Tweedie, R. L. (1974) R-theory for Markov chains on a general state space I: solidarity properties and R-recurrent chains. Ann. Probability 2, 840–864.Google Scholar
[8] Tweedie, R. L. (1975) Sufficient conditions for ergodicity of Markov chains on a general state space. Stoch. Proc. Appl. To appear.CrossRefGoogle Scholar