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Genealogical constructions and asymptotics for continuous-time Markov and continuous-state branching processes

Part of: Limit theorems Markov processes

Published online by Cambridge University Press:  01 February 2019

Thomas G. Kurtz
Thomas G. Kurtz*
Affiliation:
University of Wisconsin‒Madison
*
Departments of Mathematics and Statistics, University of Wisconsin‒Madison, 480 Lincoln Drive, Madison, WI 53706-1388, USA. Email address: kurtz@math.wisc.edu
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Abstract

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Genealogical constructions of population processes provide models which simultaneously record the forward-in-time evolution of the population size (and distribution of locations and types for models that include them) and the backward-in-time genealogies of the individuals in the population at each time t. A genealogical construction for continuous-time Markov branching processes from Kurtz and Rodrigues (2011) is described and exploited to give the normalized limit in the supercritical case. A Seneta‒Heyde norming is identified as a solution of an ordinary differential equation. The analogous results are given for continuous-state branching processes, including proofs of the normalized limits of Grey (1974) in both the supercritical and critical/subcritical cases.

Keywords

Genealogical constructionMarkov branching processsupercritical limitSeneta‒Heyde normingcontinuous-state branching process

MSC classification

Primary: 60J80: Branching processes (Galton-Watson, birth-and-death, etc.)
Secondary: 60F15: Strong theorems 60J25: Continuous-time Markov processes on general state spaces
Type
Original Article
Information
Advances in Applied Probability , Volume 50 , Issue A: Branching and Applied Probability , December 2018 , pp. 197 - 209
Copyright
Copyright © Applied Probability Trust 2018 

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