Exact simulation of the genealogical tree for a stationary branching population and application to the asymptotics of its total length
- 1 June 2021
- journal article
- research article
- Published by Cambridge University Press (CUP) in Advances in Applied Probability
- Vol. 53 (2), 537-574
- https://doi.org/10.1017/apr.2020.70
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
This publication has 25 references indexed in Scilit:
- A note on the Gromov-Hausdorff-Prokhorov distance between (locally) compact metric measure spacesElectronic Journal of Probability, 2013
- Smaller population size at the MRCA time for stationary branching processesThe Annals of Probability, 2012
- A continuum-tree-valued Markov processThe Annals of Probability, 2012
- Rayleigh processes, real trees, and root growth with re-graftingProbability Theory and Related Fields, 2005
- Probabilistic and fractal aspects of Lévy treesProbability Theory and Related Fields, 2004
- Coalescence times for the branching processAdvances in Applied Probability, 2003
- Handbook of Brownian Motion-Facts and FormulaeJournal of the American Statistical Association, 1998
- T-theory: An OverviewEuropean Journal of Combinatorics, 1996
- Decomposition du mouvement brownien avec dérive en un minimum local par juxtaposition de ses excursions positives et négativesPublished by Springer Science and Business Media LLC ,1991
- Markov FunctionsThe Annals of Probability, 1981