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

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. DOI: https://doi.org/10.1214/08-AOP441 Posted at the Zurich Open Repository and Archive, University of Zurich ZORA URL: https://doi.org/10.5167/uzh-78039 Published Version Originally published at: Bertoin, Jean (2009). The structure of the allelic partition of the total population for Galton-Watson processes with neutral mutations. The Annals of Probability, 37(4):1502-1523. DOI: https://doi.org/10.1214/08-AOP441 The Annals of Probability 2009, Vol. 37, No. 4, 1502–1523 DOI: 10.1214/08-AOP441 © Institute of Mathematical Statistics, 2009 THE STRUCTURE OF THE ALLELIC PARTITION OF THE TOTAL POPULATION FOR GALTON–WATSON PROCESSES WITH NEUTRAL MUTATIONS