Moments of the frequency spectrum of a splitting tree with neutral Poissonian mutations

TL;DR

This paper analyzes the allele frequency spectrum in a branching population modeled by splitting trees with neutral Poissonian mutations, providing recursive formulas for moments and proving convergence properties.

Contribution

It introduces a new construction of the coalescent point process to compute moments of the frequency spectrum in splitting trees with mutations.

Findings

Recursive formulas for joint factorial moments of allele counts

Elementary proof of almost sure convergence of the frequency spectrum

Analysis applicable to supercritical splitting trees

Abstract

We consider a branching population where individuals live and reproduce independently. Their lifetimes are i.i.d. and they give birth at a constant rate b. The genealogical tree spanned by this process is called a splitting tree, and the population counting process is a homogeneous, binary Crump-Mode-Jagers process. We suppose that mutations affect individuals independently at a constant rate $\\theta$ during their lifetimes, under the infinite-alleles assumption: each new mutation gives a new type, called allele, to his carrier. We study the allele frequency spectrum which is the numbers A(k, t) of types represented by k alive individuals in the population at time t. Thanks to a new construction of the coalescent point process describing the genealogy of individuals in the splitting tree, we are able to compute recursively all joint factorial moments of (A(k, t)) k$\ge$1. These moments allow us to give an elementary proof of the almost sure convergence of the frequency spectrum in a supercritical splitting tree.