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Explosive Markov branching processes: entrance laws and limiting behaviour

Part of: Markov processes

Published online by Cambridge University Press:  01 July 2016

Anthony G. Pakes
Anthony G. Pakes*
Affiliation:
University of Western Australia
*
* Postal address: Department of Mathematics, University of Western Australia, Nedlands, WA 6009, Australia.
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Abstract

The supercritical Markov branching process is examined in the case where the minimal version of the process has strictly substochastic transition laws. This provides a nice example of the general construction theory for discrete-state Markov processes.

Entrance laws corresponding to the minimal process are characterised. Limit properties of the processes constructed from these entrance laws are examined. All such processes which are honest and cannot hit zero are ergodic. Otherwise these processes are λ-positive and limit theorems conditional on not having left the positive states are given.

A connection is made with recent work on the general construction problem when a λ-subinvariant measure is given. The case where immigration is allowed is mentioned.

Keywords

MARKOV PROCESSESENTRANCE LAWSINVARIANT MEASURES

MSC classification

Primary: 60J80: Branching processes (Galton-Watson, birth-and-death, etc.)
Secondary: 60J10: Markov chains (discrete-time Markov processes on discrete state spaces)
Type
Research Article
Information
Advances in Applied Probability , Volume 25 , Issue 4 , December 1993 , pp. 737 - 756
Copyright
Copyright © Applied Probability Trust 1993 

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