The Annals of Probability

The structure of the allelic partition of the total population for Galton–Watson processes with neutral mutations

Jean Bertoin

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Abstract

We consider a (sub-)critical Galton–Watson process with neutral mutations (infinite alleles model), and decompose the entire population into clusters of individuals carrying the same allele. We specify the law of this allelic partition in terms of the distribution of the number of clone-children and the number of mutant-children of a typical individual. The approach combines an extension of Harris representation of Galton–Watson processes and a version of the ballot theorem. Some limit theorems related to the distribution of the allelic partition are also given.

Article information

Source
Ann. Probab. Volume 37, Number 4 (2009), 1502-1523.

Dates
First available in Project Euclid: 21 July 2009

Permanent link to this document
http://projecteuclid.org/euclid.aop/1248182146

Digital Object Identifier
doi:10.1214/08-AOP441

Mathematical Reviews number (MathSciNet)
MR2546753

Zentralblatt MATH identifier
1180.92063

Subjects
Primary: 60J80: Branching processes (Galton-Watson, birth-and-death, etc.) 60J10: Markov chains (discrete-time Markov processes on discrete state spaces)

Keywords
Branching process infinite alleles model allelic partition ballot theorem

Citation

Bertoin, Jean. The structure of the allelic partition of the total population for Galton–Watson processes with neutral mutations. The Annals of Probability 37 (2009), no. 4, 1502--1523. doi:10.1214/08-AOP441. http://projecteuclid.org/euclid.aop/1248182146.


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