Exact simulation of the genealogical tree for a stationary branching population and application to the asymptotics of its total length - Archive ouverte HAL Access content directly
Journal Articles Advances in Applied Probability Year : 2021

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.
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Dates and versions

hal-01413614 , version 1 (10-12-2016)
hal-01413614 , version 2 (18-04-2018)
hal-01413614 , version 3 (30-01-2020)

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Jean-François Delmas, Romain Abraham. Exact simulation of the genealogical tree for a stationary branching population and application to the asymptotics of its total length. Advances in Applied Probability, 2021, 53, pp.537-574. ⟨hal-01413614v3⟩
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