Hostname: page-component-586b7cd67f-dsjbd Total loading time: 0 Render date: 2024-11-25T05:08:32.782Z Has data issue: false hasContentIssue false
  • English
  • Français

Coalescence in Critical and Subcritical Galton-Watson Branching Processes

Part of: Limit theorems Markov processes

Published online by Cambridge University Press:  04 February 2016

K. B. Athreya
K. B. Athreya*
Affiliation:
Iowa State University
*
∗ Postal address: Departments of Mathematics and Statistics, Iowa State University, Ames, Iowa, 50011, USA. Email address: [email protected]
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

In a Galton-Watson branching process that is not extinct by the nth generation and has at least two individuals, pick two individuals at random by simple random sampling without replacement. Trace their lines of descent back in time till they meet. Call that generation Xn a pairwise coalescence time. Similarly, let Yn denote the coalescence time for the whole population of the nth generation conditioned on the event that it is not extinct. In this paper the distributions of Xn and Yn, and their limit behaviors as n → ∞ are discussed for both the critical and subcritical cases.

Keywords

Branching processcoalescencecriticalsubcritical

MSC classification

Primary: 60J80: Branching processes (Galton-Watson, birth-and-death, etc.)
Secondary: 60F10: Large deviations
Type
Research Article
Information
Journal of Applied Probability , Volume 49 , Issue 3 , September 2012 , pp. 627 - 638
Copyright
© Applied Probability Trust 

References

Athreya, K. B. (2012). Coalescence in the recent past in rapidly growing populations. Stoch. Process. Appl. 122, 3757–3766.Google Scholar
Athreya, K. B. and Ney, P. E. (2004). Branching Processes. Dover, Mineola, NY.Google Scholar
Geiger, J. (1999). Elementary new proofs of classical limit theorems for Galton–Watson processes. J. Appl. Prob. 36, 301–309.Google Scholar
Kallenberg, O. (1986). Random Measures, 4th edn. Academic Press, London.Google Scholar
Le Gall, J.-F. (2010). Itô's excursion theory and random trees. Stoch. Process. Appl. 120, 712–749.Google Scholar
Zubkov, A. M. (1975). Limit distribution of the distance to the nearest common ancestor. Theory Prob. Appl. 20, 602–612.CrossRefGoogle Scholar