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Rank-dependent Galton‒Watson processes and their pathwise duals

Part of: Markov processes

Published online by Cambridge University Press:  01 February 2019

Serik Sagitov  and
Jonas Jagers
Serik Sagitov*
Affiliation:
Chalmers University of Technology and University of Gothenburg
Jonas Jagers*
Affiliation:
Chalmers University of Technology
*
Department of Mathematical Sciences, Chalmers University of Technology, 412 96 Gothenburg, Sweden. Email address: [email protected]
Department of Mathematical Sciences, Chalmers University of Technology, 412 96 Gothenburg, Sweden. Email address: [email protected]
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Abstract

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We introduce a modified Galton‒Watson process using the framework of an infinite system of particles labelled by (x,t), where x is the rank of the particle born at time t. The key assumption concerning the offspring numbers of different particles is that they are independent, but their distributions may depend on the particle label (x,t). For the associated system of coupled monotone Markov chains, we address the issue of pathwise duality elucidated by a remarkable graphical representation in which the trajectories of the primary Markov chains and their duals coalesce to form forest graphs on a two-dimensional grid.

Keywords

Graphical representationSiegmund's dualitydual forestlinear-fractional reproductionbirth‒death process

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
Original Article
Information
Advances in Applied Probability , Volume 50 , Issue A: Branching and Applied Probability , December 2018 , pp. 229 - 239
Copyright
Copyright © Applied Probability Trust 2018 

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