# Approximations of the allelic frequency spectrum in general supercritical branching populations

## TL;DR

This paper develops approximation methods for the allelic frequency spectrum in general supercritical branching populations, analyzing their asymptotic errors and applying the results to genetic data interpretation.

## Contribution

It introduces new approximation techniques for the allelic frequency spectrum in complex branching models and evaluates their accuracy through asymptotic analysis.

## Key findings

- Derived central limit theorems for approximation errors
- Provided numerical analysis of approximation performance
- Applied methods to genetic data analysis, such as EHH

## 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].