Advances in Applied Probability · 1976 · 43 citations · 15 references
EngineeringMathematicsProbability TheorySet TheoryStochastic ProcessesMarkov ChainsMarkov ProcessesChain RecurrenceContinuous-time Markov ChainsCountable-state Markov ChainsFinite-state Markov ChainsNull RecurrencePositive RecurrenceRecurrenceRecurrence Properties
When a Markov chain is irreducible, criteria for classifying the chain itself can be derived from the status of the sets it visits, and the paper discusses how these status sets relate to recurrence, transience, nullity, and positivity. The paper aims to present a comprehensive set of criteria for classifying the sets visited by a general state‑space Markov chain as recurrent, transient, null, or positive. To achieve this, the authors first establish connections between various definitions of recurrence, transience, nullity, and positivity for sets and for irreducible chains, then formulate explicit criteria for classifying sets. The resulting criteria specialize to Foster’s classical results for countable state spaces, extend Lamperti’s work for continuous‑valued chains, and strengthen Tweedie’s positivity and recurrence conditions for general spaces.
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).
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