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 θ 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≥1. These moments allow us to give an elementary proof of the almost sure convergence of the frequency spectrum in a supercritical splitting tree.
Highlights
In this work, we study a branching population in which every individual is supposed to have a lifetime independent from the other individuals in the population
The genealogical tree underlying the history of the population, the so called splitting tree, has been widely studied in the past [19, 10, 9]
We introduce a new construction of the coalescent point process
Summary
We study a branching population in which every individual is supposed to have a lifetime independent from the other individuals in the population. We prove the almost sure convergence of the frequency spectrum avoiding the use of the theory of general branching processes counted by random characteristics in the supercritical case. These moment formulas can provide many valuable informations, for instance, on the error in the aforementioned convergence, which suggest CLT-type results. Even if this result is used as a tool for this work, it is interesting by itself. Ck E , almost surely, k ≥ 1, where E is an exponential random variable with parameter 1 conditionally on non-extinction, and the constants ck are explicit
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