Hostname: page-component-cd9895bd7-gvvz8 Total loading time: 0 Render date: 2024-12-28T02:37:41.225Z Has data issue: false hasContentIssue false
  • English
  • Français

On a coalescence process and its branching genealogy

Part of: Markov processes Stochastic processes

Published online by Cambridge University Press:  09 December 2016

Nicolas Grosjean  and
Thierry Huillet
Nicolas Grosjean*
Affiliation:
Université de Cergy-Pontoise
Thierry Huillet*
Affiliation:
Université de Cergy-Pontoise
*
* Postal address: Laboratoire de Physique Théorique et Modélisation, CNRS-UMR 8089, Université de Cergy-Pontoise, 2 Avenue Adolphe Chauvin, Cergy-Pontoise, 95302, France.
* Postal address: Laboratoire de Physique Théorique et Modélisation, CNRS-UMR 8089, Université de Cergy-Pontoise, 2 Avenue Adolphe Chauvin, Cergy-Pontoise, 95302, France.
Get access Rights & Permissions [Opens in a new window]

Abstract

We define and analyze a coalescent process as a recursive box-filling process whose genealogy is given by an ancestral time-reversed, time-inhomogeneous Bienyamé‒Galton‒Watson process. Special interest is on the expected size of a typical box and its probability of being empty. Special cases leading to exact asymptotic computations are investigated when the coalescing mechanisms are either linear fractional or quadratic.

Keywords

Inhomogeneous Bienyamé‒Galton‒Watson processcoalescence processgenealogy

MSC classification

Primary: 60J80: Branching processes (Galton-Watson, birth-and-death, etc.)
Secondary: 60G20: Generalized stochastic processes
Type
Research Papers
Information
Journal of Applied Probability , Volume 53 , Issue 4 , December 2016 , pp. 1156 - 1165
Copyright
Copyright © Applied Probability Trust 2016 

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

[1] Athreya, K. B. (2012).Coalescence in critical and subcritical Galton‒Watson branching processes.J. Appl. Prob. 49,627–638.Google Scholar
[2] Athreya, K. B. (2012).Coalescence in the recent past in rapidly growing populations.Stoch. Process. Appl. 122,3757–3766.Google Scholar
[3] Athreya, K. B. and Ney, P. E. (1972).Branching Processes.Springer,New York.Google Scholar
[4] Esty, W. W. (1975).The reverse Galton‒Watson process.J. Appl. Prob. 12,574–580.Google Scholar
[5] Harris, T. E. (1963).The Theory of Branching Processes.Springer,Berlin.Google Scholar
[6] Jagers, P. (1974).Galton‒Watson processes in varying environments.J. Appl. Prob. 11,174–178.Google Scholar
[7] Lambert, A. (2003).Coalescence times for the branching process.Adv. Appl. Prob. 35,1071–1089.CrossRefGoogle Scholar
[8] Sagitov, S. and Lindo, A. (2016).A special family of Galton‒Watson processes with explosions. In Branching Processes and Their Applications,Springer,Berlin.Google Scholar
[9] Sheth, R. K. (1996).Galton‒Watson branching processes and the growth of gravitational clustering.Mon. Not. R. Astron. Soc. 281,1277–1289.Google Scholar
[10] Tuljapurkar, S. (1990).Population Dynamics in Variable Environments.Springer,Berlin.Google Scholar