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

Abstract : We consider a model of 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 of 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.
Type de document :
Pré-publication, Document de travail
2018
Domaine :

Littérature citée [23 références]

https://hal.archives-ouvertes.fr/hal-01413614
Contributeur : Romain Abraham <>
Soumis le : mercredi 18 avril 2018 - 11:17:29
Dernière modification le : jeudi 17 janvier 2019 - 14:38:04

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simul-2018-03.pdf
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### Identifiants

• HAL Id : hal-01413614, version 2
• ARXIV : 1612.03715

### Citation

Romain Abraham, Jean-François Delmas. Exact simulation of the genealogical tree for a stationary branching population and application to the asymptotics of its total length. 2018. 〈hal-01413614v2〉

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