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

TL;DR

This paper analyzes the structure of allelic partitions in subcritical Galton-Watson processes with neutral mutations, providing explicit laws and limit theorems for the distribution of alleles within the population.

Contribution

It introduces a novel decomposition of the population into allelic clusters using an extension of Harris representation and the ballot theorem, with new limit theorems for allele distributions.

Findings

Explicit law for allelic partition in Galton-Watson processes

Limit theorems for allele distribution in subcritical regimes

Extension of Harris representation and ballot theorem methods

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.