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Exact simulation of the genealogical tree for a stationary branching population and application to the asymptotics of its total length

Part of: Markov processes

Published online by Cambridge University Press:  01 July 2021

Romain Abraham  and
Jean-François Delmas
Romain Abraham*
Affiliation:
IDP, Université d’Orléans, Université de Tours, CNRS
Jean-François Delmas*
Affiliation:
CERMICS, Université Paris-Est, ENPC
*
*Postal address: Institut Denis Poisson, Université d’Orléans, B.P. 6759, 45067 Orléans Cedex 2, France. E-mail: romain.abraham@univ-orleans.fr
**Postal address: école des Ponts ParisTech, CERMICS, 6 et 8, avenue Blaise Pascal, Cité Descartes—Champs-sur-Marne, 77455 Marne-la-Vallée Cedex 2, France.
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Abstract

We consider a model of a stationary population with random size given by a continuous-state branching process with immigration with a quadratic branching mechanism. We give an exact elementary simulation procedure for the genealogical tree of n individuals randomly chosen among the extant population at a given time. Then we prove the convergence of the renormalized total length of this genealogical tree as n goes to infinity; see also Pfaffelhuber, Wakolbinger and Weisshaupt (2011) in the context of a constant-size population. The limit appears already in Bi and Delmas (2016) but with a different approximation of the full genealogical tree. The proof is based on the ancestral process of the extant population at a fixed time, which was defined by Aldous and Popovic (2005) in the critical case.

Keywords

Stationary branching processesreal treesgenealogical treesancestral processsimulation

MSC classification

Primary: 60J80: Branching processes (Galton-Watson, birth-and-death, etc.)
Secondary: 60J85: Applications of branching processes
Type
Original Article
Information
Advances in Applied Probability , Volume 53 , Issue 2 , June 2021 , pp. 537 - 574
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Copyright
© The Author(s), 2021. Published by Cambridge University Press on behalf of Applied Probability Trust

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