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Criteria for classifying general Markov chains

Published online by Cambridge University Press:  01 July 2016

R. L. Tweedie
R. L. Tweedie*
Affiliation:
C.S.I.R.O. Division of Mathematics and Statistics, Canberra
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Abstract

The aim of this paper is to present a comprehensive set of criteria for classifying as recurrent, transient, null or positive the sets visited by a general state space Markov chain. When the chain is irreducible in some sense, these then provide criteria for classifying the chain itself, provided the sets considered actually reflect the status of the chain as a whole. The first part of the paper is concerned with the connections between various definitions of recurrence, transience, nullity and positivity for sets and for irreducible chains; here we also elaborate the idea of status sets for irreducible chains. In the second part we give our criteria for classifying sets. When the state space is countable, our results for recurrence, transience and positivity reduce to the classical work of Foster (1953); for continuous-valued chains they extend results of Lamperti (1960), (1963); for general spaces the positivity and recurrence criteria strengthen those of Tweedie (1975b).

Keywords

MARKOV CHAINRECURRENCETRANSIENCEPOSITIVE RECURRENCEERGODICITYNULL RECURRENCESTATIONARY MEASUREINVARIANT MEASUREIRREDUCIBILITYSTATUS SETMEAN DRIFT
Type
Research Article
Information
Advances in Applied Probability , Volume 8 , Issue 4 , December 1976 , pp. 737 - 771
Copyright
Copyright © Applied Probability Trust 1976 

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References

References added in proof

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