[hal-00191132, v3] The structure of the allelic partition of the total population for Galton-Watson
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The Annals of Probability2009, Vol. 37, No. 4, 1502–1523DOI: 10.1214/08-AOP441c Institute of Mathematical Statistics, 2009THE STRUCTURE OF THE ALLELIC PARTITION OFTHE TOTAL POPULATION FOR GALTON–WATSONPROCESSES WITH NEUTRAL MUTATIONSBy Jean BertoinUniversit´e Pierre et Marie CurieWe consider a (sub-)critical Galton–Watson process with neutralmutations (infinite alleles model), and decompose the entire popula-tion into clusters of individuals carrying the same allele. We specifythelawofthisallelic partitionintermsofthedistributionofthenum-ber of clone-children and the number of mutant-children of a typicalindividual.TheapproachcombinesanextensionofHarris representa-tion of Galton–Watson processes and a version of the ballot theorem.Somelimittheorems relatedtothedistributionof theallelic partitionare also given.1. Introduction. We consider a Galton–Watson process, that is, a popu-lation model with asexual reproduction such that at every generation, eachindividual gives birth to a random number of children according to a fixeddistribution and independently of the other individuals in the population.We are interested in the situation where a child can be either a clone, thatis, of the same type (or allele) as its parent, or a mutant, that is, of a newtype. We stress that each mutant has a distinct type and in turn gives birthto clones of itself and to new mutants according to the same statistical lawas its parent, even though it bears a different ...
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