The coalescent of a sample from a linear-fractional branching process

TL;DR

This paper investigates the genealogical structures of samples from linear-fractional Bienaymé-Galton-Watson processes under different sampling schemes, revealing relationships between their resulting trees.

Contribution

It provides a novel analysis of the genealogical trees resulting from Bernoulli and uniform sampling in linear-fractional branching processes.

Findings

Relationship established between tree distributions from different sampling methods

Analytical characterization of sampled genealogical trees

Insights into sampling effects on genealogical structures

Abstract

In this article, we focus on Bienaym\'e-Galton-Watson processes with linear-fractional offspring distributions. At a fixed generation, we consider a sample of the individuals alive, drawn in two different ways: either through Bernoulli sampling, where each individual is selected independently with a given probability, or through uniform sampling, where a fixed number of individuals are chosen uniformly at random. We analyze the genealogical trees generated by the sampled individuals. In particular, we establish a relationship between the distributions of the trees resulting from the two sampling schemes.