Hostname: page-component-74656f9884-j7cmm Total loading time: 0 Render date: 2026-11-10T13:51:30.915Z Has data issue: false hasContentIssue false
  • English
  • Français

Coalescence times for the branching process

Part of: Markov processes

Published online by Cambridge University Press:  01 July 2016

Amaury Lambert
Amaury Lambert*
Affiliation:
Ecole Normale Supérieure, Paris
*
∗ Postal address: Unit of Mathematical Evolutionary Biology, Fonctionnement et Evolution des Systèmes Ecologiques UMR 7625, Ecole Normale Supérieure, 46, rue d'Ulm, F-75230 Paris Cedex 05, France. Email address: alambert@ens.fr
Get access Rights & Permissions [Opens in a new window]

Abstract

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 branching process founded t units of time ago, in both the discrete and continuous (time and state-space) settings. We obtain limiting distributions as t→∞ in the subcritical case. In the continuous setting, these distributions are specified for quadratic branching mechanisms (corresponding to Brownian motion and Brownian motion with positive drift), and we also 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

GenealogyBienaymé-Galton-Watson processcontinuous-state branching processcoalescencequasi-stationary distribution

MSC classification

Primary: 60J80: Branching processes (Galton-Watson, birth-and-death, etc.)

Information

Type
General Applied Probability
Information
Advances in Applied Probability , Volume 35 , Issue 4 , December 2003 , pp. 1071 - 1089
Copyright
Copyright © Applied Probability Trust 2003 

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.)

Article purchase

Temporarily unavailable

References

[1] Aldous, D. J. (1999). Deterministic and stochastic models for coalescence (aggregation and coagulation): a review of the mean-field theory for probabilists. Bernoulli 5, 3–48.Google Scholar
[2] Athreya, K. B. and Ney, P. E. (1972). Branching Processes. Springer, New York.Google Scholar
[3] Bertoin, J. and Le Gall, J. F. (2000). The Bolthausen–Sznitman coalescent and the genealogy of continuous-state branching processes. Prob. Theory Relat. Fields 117, 249–266.Google Scholar
[4] Bingham, N. H. (1976). Continuous branching processes and spectral positivity. Stoch. Process. Appl. 4, 217–242.Google Scholar
[5] Bolthausen, E. and Sznitman, A. S. (1998). On Ruelle's probability cascades and an abstract cavity method. Commun. Math. Phys. 197, 247–276.Google Scholar
[6] Evans, S. N. and Pitman, J. (1998). Construction of Markovian coalescents. Ann. Inst. H. Poincaré Prob. Statist. 34, 339–383.CrossRefGoogle Scholar
[7] Grey, D. R. (1974). Asymptotic behaviour of continuous-time, continuous state-space branching processes. J. Appl. Prob. 11, 669–677.CrossRefGoogle Scholar
[8] Heathcote, C. R., Seneta, E. and Vere-Jones, D. (1967). A refinement of two theorems in the theory of branching processes. Teor. Veroyat. Primen. 12, 341–346.Google Scholar
[9] Kingman, J. F. C. (1982). The coalescent. Stoch. Process. Appl. 13, 235–248.Google Scholar
[10] Kolmogorov, A. N. (1938). Zur Lösung einer biologischen Aufgabe. Commun. Math. Mech. Chebyshev Univ. Tomsk 2, 1–6.Google Scholar
[11] Lambert, A. (2001). Arbres, excursions et processus de Lévy complètement asymétriques. Doctoral Thesis, Université Pierre et Marie Curie, Paris.Google Scholar
[12] Lambert, A. (2002). The genealogy of continuous-state branching processes with immigration. Prob. Theory Relat. Fields 122, 42–70.CrossRefGoogle Scholar
[13] Lamperti, J. (1967). Continuous-state branching processes. Bull. Am. Math. Soc. 73, 382–386.Google Scholar
[14] Le Gall, J. F. and Le Jan, Y. (1998). Branching processes in Lévy processes: the exploration process. Ann. Prob. 26, 213–252.Google Scholar
[15] Lyons, R., Pemantle, R. and Peres, Y. (1995). Conceptual proofs of L log L criteria for mean behavior of branching processes. Ann. Prob. 23, 1125–1138.Google Scholar
[16] Möhle, M. and Sagitov, S. (2001). A classification of coalescent processes for haploid exchangeable population models. Ann. Prob. 29, 1547–1562.Google Scholar
[17] O'Connell, N. (1995). The genealogy of branching processes and the age of our most recent common ancestor. Adv. Appl. Prob. 27, 418–442.Google Scholar
[18] Schweinsberg, J. (2003). Coalescent processes obtained from supercritical Galton–Watson processes. Stoch. Process. Appl. 106, 107–139.Google Scholar
[19] Seneta, E. and Vere-Jones, D. (1966). On quasi-stationary distributions in discrete-time Markov chains with a denumerable infinity of states. J. Appl. Prob. 3, 403–434.Google Scholar
[20] Yaglom, A. M. (1947). Certain limit theorems of the theory of branching stochastic processes. Dokl. Akad. Nauk. SSSR 56, 795–798 (in Russian).Google Scholar