Non-explosivity of limits of conditioned birth and death processes

Roberts, G.O., Jacka, S.D. and Pollett, P.K.

Abstract: Let X be a birth and death process on with absorption at zero and suppose that X is suitably recurrent, irreducible, and non-explosive. In a recent paper, Roberts and Jacka (1994) showed that as the process conditioned to non-absorption until time T converges weakly to a time-homogeneous Markov limit, , which is itself a birth and death process. However the question of the possibility of explosiveness of remained open. The major result of this paper establishes that is always non-explosive.

Keywords: Invariant measures, quasi-stationary distributions.

Acknowledgements: The research of P. Pollett was supported by the Australia Research Council (Grant No. A69130032).

The authors:

• Gareth Roberts, Statistical Laboratory, University of Cambridge. G.O.Roberts@statslab.cam.ac.uk
• Saul Jacka, Department of Statistics, University of Warwick. S.D.Jacka@warwick.ac.uk.
• Phil Pollett, Discipline of Mathematics, The University of Queensland. pkp@maths.uq.edu.au

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Last modified: 21st March 1998