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The Limit Behavior of Dual Markov Branching Processes

Part of: Markov processes

Published online by Cambridge University Press:  14 July 2016

Yangrong Li ,
Anthony G. Pakes ,
Jia Li  and
Anhui Gu
Yangrong Li*
Affiliation:
Southwest China University and Institute of Applied Physics and Computational Mathematics, Beijing
Anthony G. Pakes*
Affiliation:
University of Western Australia
Jia Li*
Affiliation:
Southwest China University
Anhui Gu*
Affiliation:
Southwest China University
*
∗Postal address: School of Mathematics and Statistics, Southwest China University, Chongqing, 400715, People's Republic of China.
∗∗∗Postal address: School of Mathematics and Statistics, University of Western Australia, 35 Stirling Highway, Crawley, WA 6009, Australia. Email address: pakes@maths.uwa.edu.au
∗Postal address: School of Mathematics and Statistics, Southwest China University, Chongqing, 400715, People's Republic of China.
∗Postal address: School of Mathematics and Statistics, Southwest China University, Chongqing, 400715, People's Republic of China.
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Abstract

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A dual Markov branching process (DMBP) is by definition a Siegmund's predual of some Markov branching process (MBP). Such a process does exist and is uniquely determined by the so-called dual-branching property. Its q-matrix Q is derived and proved to be regular and monotone. Several equivalent definitions for a DMBP are given. The criteria for transience, positive recurrence, strong ergodicity, and the Feller property are established. The invariant distributions are given by a clear formulation with a geometric limit law.

Keywords

Markov branching processdual Markov branching processdual-branching q-matrixtransiencerecurrencestrong ergodicityinvariant distribution

MSC classification

Secondary: 60J27: Continuous-time Markov processes on discrete state spaces 60J80: Branching processes (Galton-Watson, birth-and-death, etc.)
Type
Research Article
Information
Journal of Applied Probability , Volume 45 , Issue 1 , March 2008 , pp. 176 - 189
Copyright
Copyright © Applied Probability Trust 2008 

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