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

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...

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Bibliographic Details
Published in:Advances in applied probability 2021-06, Vol.53 (2), p.537-574
Main Authors: Abraham, Romain, Delmas, Jean-François
Format: Article
Language:English
Subjects:
Immigration
Mathematics
Mitochondrial DNA
Original Article
Original Articles
Population
Probability
Random variables
Simulation
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Summary: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.
ISSN:0001-8678
1475-6064
DOI:10.1017/apr.2020.70