A limit theorem for trees of alleles in branching processes with rare neutral mutations

TL;DR

This paper proves a limit theorem describing the genealogical structure of alleles in large, neutrally mutating populations modeled by branching processes, showing convergence to a continuous state branching process.

Contribution

It establishes a new limit theorem for allele size distributions in branching processes with rare neutral mutations, linking genealogical structures to continuous state processes.

Findings

Allele size processes converge to a Jirina process in the specified regime.

Utilizes Ito's excursion theory and Levy-Itô decomposition for analysis.

Provides a mathematical framework for understanding genetic diversity in large populations.

Abstract

We are interested in the genealogical structure of alleles for a Bienaym\'e-Galton-Watson branching process with neutral mutations (infinite alleles model), in the situation where the initial population is large and the mutation rate small. We shall establish that for an appropriate regime, the process of the sizes of the allelic sub-families converges in distribution to a certain continuous state branching process (i.e. a Jirina process) in discrete time. It\^o's excursion theory and the L\'eevy-It\^o decomposition of subordinators provide fundamental insights for the results.