Title: Approximations of the allelic frequency spectrum in general supercritical branching populations,
Abstract: We consider a general branching population where the lifetimes of individuals
are i.i.d.\ with arbitrary distribution and where each individual gives birth
to new individuals at Poisson times independently from each other. In addition,
we suppose that individuals experience mutations at Poissonian rate $\theta$
under the infinitely many alleles assumption assuming that types are
transmitted from parents to offspring. This mechanism leads to a partition of
the population by type, called the allelic partition. The main object of this
work is the frequency spectrum $A(k,t)$ which counts the number of families of
size $k$ in the population at time $t$. The process $(A(k,t),\
t\in\mathbb{R}_+)$ is an example of non-Markovian branching process belonging
to the class of general branching processes counted by random characteristics.
In this work, we propose methods of approximation to replace the frequency
spectrum by simpler quantities. Our main goal is study the asymptotic error
made during these approximations through central limit theorems. In a last
section, we perform several numerical analysis using this model, in particular
to analyze the behavior of one of these approximations with respect to Sabeti's
Extended Haplotype Homozygosity [18].