Hostname: page-component-cd9895bd7-mkpzs Total loading time: 0 Render date: 2024-12-23T13:50:43.707Z Has data issue: false hasContentIssue false
  • English
  • Français

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

Part of: Markov processes

Published online by Cambridge University Press:  01 July 2021

Romain Abraham  and
Jean-François Delmas
Romain Abraham*
Affiliation:
IDP, Université d’Orléans, Université de Tours, CNRS
Jean-François Delmas*
Affiliation:
CERMICS, Université Paris-Est, ENPC
*
*Postal address: Institut Denis Poisson, Université d’Orléans, B.P. 6759, 45067 Orléans Cedex 2, France. E-mail: [email protected]
**Postal address: école des Ponts ParisTech, CERMICS, 6 et 8, avenue Blaise Pascal, Cité Descartes—Champs-sur-Marne, 77455 Marne-la-Vallée Cedex 2, France.
Get access Rights & Permissions [Opens in a new window]

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

Stationary branching processesreal treesgenealogical treesancestral processsimulation

MSC classification

Primary: 60J80: Branching processes (Galton-Watson, birth-and-death, etc.)
Secondary: 60J85: Applications of branching processes
Type
Original Article
Information
Advances in Applied Probability , Volume 53 , Issue 2 , June 2021 , pp. 537 - 574
Check for updates
Copyright
© The Author(s), 2021. Published by Cambridge University Press on behalf of Applied Probability Trust

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Abraham, R. and Delmas, J.-F. (2009). Williams’ decomposition of the Lévy continuum random tree and simultaneous extinction probability for populations with neutral mutations. Stoch. Process. Appl. 119, 1124–1143.CrossRefGoogle Scholar
Abraham, R. and Delmas, J.-F. (2012). A continuum-tree-valued Markov process. Ann. Prob. 40, 1167–1211.CrossRefGoogle Scholar
Abraham, R. and Delmas, J.-F. (2018). Reversal property of the Brownian tree. ALEA Latin Amer. J. Prob. Math. Statist. 15, 1293–1309.CrossRefGoogle Scholar
Abraham, R., Delmas, J.-F. and Hoscheit, P. (2013). A note on the Gromov–Hausdorff–Prokhorov distance between (locally) compact metric measure spaces. Electron. J. Prob. 18, 21.CrossRefGoogle Scholar
Aldous, D. (1991). The continuum random tree. I. Ann. Prob. 19, 1–28.Google Scholar
Aldous, D. and Popovic, L. (2005). A critical branching process model for biodiversity. Adv. Appl. Prob. 37, 1094–1115.CrossRefGoogle Scholar
Athreya, S., Löhr, W. and Winter, A. (2017). Invariance principle for variable speed random walks on trees. Ann. Prob. 45, 625–667.CrossRefGoogle Scholar
Berestycki, J., Berestycki, N. and Limic, V. (2010). The $\Lambda$-coalescent speed of coming down from infinity. Ann. Prob. 38, 207–233.CrossRefGoogle Scholar
Bertoin, J. (1991). Décomposition du mouvement brownien avec dérive en un minimum local par juxtaposition de ses excursions positives et négatives. In Séminaire de Probabilités, XXV, Springer, Berlin, pp. 330–344.CrossRefGoogle Scholar
Bertoin, J. (1996). Lévy Processes. Cambridge University Press.Google Scholar
Bi, H. and Delmas, J.-F. (2016). Total length of the genealogical tree for quadratic stationary continuous-state branching processes. Ann. Inst. H. Poincaré Prob. Statist. 52, 1321–1350.CrossRefGoogle Scholar
Borodin, A. N. and Salminen, P. (2002). Handbook of Brownian motion—Facts and Formulae, 2nd edn. Birkhäuser, Basel.CrossRefGoogle Scholar
Burago, D., Burago, Y. D. and Ivanov, S. (2001). A Course in Metric Geometry. American Mathematical Society, Providence, RI.CrossRefGoogle Scholar
Chen, Y.-T. and Delmas, J.-F. (2012). Smaller population size at the MRCA time for stationary branching processes. Ann. Prob. 40, 2034–2068.CrossRefGoogle Scholar
Dress, A., Moulton, V. and Terhalle, W. (1996). T-theory: an overview. Europ. J. Combinatorics 17, 161–175.CrossRefGoogle Scholar
Duquesne, T. (2006). The coding of compact real trees by real valued functions. Preprint. Available at https://arxiv.org/abs/math/0604106.Google Scholar
Duquesne, T. (2009). Continuum random trees and branching processes with immigration. Stoch. Process. Appl. 119, 99–129.CrossRefGoogle Scholar
Duquesne, T. and Le Gall, J.-F. (2002). Random trees, Lévy processes and spatial branching processes. Astérisque 281, 153 pp.Google Scholar
Duquesne, T. and Le Gall, J.-F. (2005). Probabilistic and fractal aspects of Lévy trees. Prob. Theory Relat. Fields 131, 553–603.CrossRefGoogle Scholar
Evans, S. N. (2008). Probability and Real Trees. Springer, Berlin.CrossRefGoogle Scholar
Evans, S. N., Pitman, J. and Winter, A. (2006). Rayleigh processes, real trees, and root growth with re-grafting. Prob. Theory Relat. Fields 134, 81–126.CrossRefGoogle Scholar
Lambert, A. (2003). Coalescence times for the branching process. Adv. Appl. Prob. 35, 1071–1089.CrossRefGoogle Scholar
Lambert, A. (2007). Quasi-stationary distributions and the continuous-state branching process conditioned to be never extinct. Electron. J. Prob. 12, 420–446.CrossRefGoogle Scholar
Le Gall, J.-F. and Le Jan, Y. (1998). Branching processes in Lévy processes: the exploration process. Ann. Prob. 26, 213–252.Google Scholar
Pfaffelhuber, P., Wakolbinger, A. and Weisshaupt, H. (2011). The tree length of an evolving coalescent. Prob. Theory Relat. Fields 151, 529–557.CrossRefGoogle Scholar
Popovic, L. (2004). Asymptotic genealogy of a critical branching process. Ann. Appl. Prob. 14, 2120–2148.CrossRefGoogle Scholar
Roelly-Coppoletta, S. and Rouault, A. (1989). Processus de Dawson–Watanabe conditionné par le futur lointain. C. R. Acad. Sci. Paris 309, 867–872.Google Scholar
Rogers, L. C. G. and Pitman, J. W. (1981). Markov functions. Ann. Prob. 9, 573–582.CrossRefGoogle Scholar
Sainudiin, R., Thatte, B. and Véber, A. (2016). Ancestries of a recombining diploid population. J. Math. Biol. 72, 363–408.CrossRefGoogle ScholarPubMed