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Semi-Markov processes on a general state space: α-theory and quasi-stationarity

Part of: Special processes

Published online by Cambridge University Press:  09 April 2009

E. Arjas ,
E. Nummelin  and
R. L. Tweedie
E. Arjas
Affiliation:
Department of Mathematics, University of British Columbia, 2075 Wesbrook Mall, Vancouver, Canada
E. Nummelin
Affiliation:
Institute of Mathematics, Helsinki University of Technology, SF-02150, Otaniemi, Finland
R. L. Tweedie
Affiliation:
Division of Mathematics and Statistics, C.S.I.R.O., P. O. Box 310, South Melbourne, Australia 3205
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Abstract

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By amalgamating the approaches of Tweedie (1974) and Nummelin (1977), an α-theory is developed for general semi-Markov processes. It is shown that α-transient, α-recurrent and α-positive recurrent processes can be defined, with properties analogous to those for transient, recurrent and positive recurrent processes. Limit theorems for α-positive recurrent processes follow by transforming to the probabilistic case, as in the above references: these then give results on the existence and form of quasistationary distributions, extending those of Tweedie (1975) and Nummelin (1976).

MSC classification

Secondary: 60K15: Markov renewal processes, semi-Markov processes
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 30 , Issue 2 , December 1980 , pp. 187 - 200
Copyright
Copyright © Australian Mathematical Society 1980

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