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Coalescence Times for the Bienaymé-Galton-Watson Process

Part of: Markov processes

Published online by Cambridge University Press:  30 January 2018

V. Le
V. Le*
Affiliation:
Université de Provence
*
∗ Postal address: Laboratoire d'Analyse Topologie et Probabilités (LATP/UMR 7353), Université d'Aix-Marseille, 39 rue F. Joliot-Curie, F-13453 Marseille cedex 13, France. Email address: levi121286@gmail.com
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Abstract

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We investigate the distribution of the coalescence time (most recent common ancestor) for two individuals picked at random (uniformly) in the current generation of a continuous-time Bienaymé-Galton-Watson process founded t units of time ago. We also obtain limiting distributions as t → ∞ in the subcritical case. We extend our results for two individuals to the joint distribution of coalescence times for any finite number of individuals sampled in the current generation.

Keywords

Bienaymé-Galton-Watson processcoalescencediscrete-state branching processquasistationary distribution

MSC classification

Primary: 60J27: Continuous-time Markov processes on discrete state spaces
Secondary: 60J80: Branching processes (Galton-Watson, birth-and-death, etc.)
Type
Research Article
Information
Journal of Applied Probability , Volume 51 , Issue 1 , March 2014 , pp. 209 - 218
Copyright
© Applied Probability Trust 

References

Athreya, K. B. and Ney, P. E. (1972). Branching Processes. Springer, New York.CrossRefGoogle Scholar
Cattiaux, P. et al. (2009). Quasi-stationary distributions and diffusion models in population dynamics. Ann. Prob. 37, 1926–1926.CrossRefGoogle Scholar
Cheong, C. K. (1970). Quasi-stationary distributions in semi-Markov processes. J. Appl. Prob. 7, 388–399, 788.Google Scholar
Cheong, C. K. (1972). Quasi-stationary distributions for the continuous-time Galton–Watson process. Bull. Soc. Math. Belg. 24, 343–350.Google Scholar
Jagers, P., Klebaner, F. and Sagitov, S. (2007). Markovian paths to extinction. Adv. Appl. Prob. 39, 569–587.Google Scholar
Joffe, A. (1967). On the Galton–Watson branching processes with mean less than one. Ann. Math. Statist. 38, 264–266.Google Scholar
Lambert, A. (2003). Coalescence times for the branching process. Adv. Appl. Prob. 35, 1071–1089.Google 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
Lambert, A. (2009). The allelic partition for coalescent point processes. Markov Process. Relat. Fields 15, 359–386.Google Scholar
Lambert, A. and Popovic, L. (2013). The coalescent point process of branching trees. Ann. Appl. Prob. 23, 99–144.CrossRefGoogle Scholar
Méléard, S. and Villemonais, D. (2012). Quasi-stationary distributions for population processes. Prob. Surveys 9, 340–410.CrossRefGoogle Scholar
Pardoux, E. (2014). Probabilistic Models of Population Genetics. In preparation.Google Scholar
Pfaffelhuber, P. and Wakolbinger, A. (2006). The process of most recent common ancestors in an evolving coalescent. Stoch. Process. Appl. 116, 1836–1859.CrossRefGoogle Scholar
Pfaffelhuber, P., Wakolbinger, A. and Weisshaupt, H. (2011). The tree length of an evolving coalescent. Prob. Theory Relat. Fields 151, 529–557.Google Scholar
Vere-Jones, D. (1969). Some limit theorems for evanescent processes. Austral. J. Statist. 11, 67–78.CrossRefGoogle Scholar
Zolotarev, V. M. (1957). More exact statements of several theorems in the theory of branching processes. Theory Prob. Appl. 2, 245–253.Google Scholar