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

Romain Abraham (1) , Jean-François Delmas (2)

1 IDP - Institut Denis Poisson
2 CERMICS - Centre d'Enseignement et de Recherche en Mathématiques et Calcul Scientifique

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.

Domaines: Probabilités [math.PR]

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⟩