On branching process with rare neutral mutation

TL;DR

This paper investigates the genealogical structure of a Galton-Watson process with neutral mutations, extending existing models to include conditioned processes and convergence results under various variance conditions.

Contribution

It extends Bertoin's work by constructing conditioned models and proving convergence of allelic sub-populations under different variance assumptions.

Findings

Constructed non-extinct conditioned models in the critical case.

Proved convergence of allelic sub-populations to a CSBP with immigration.

Established limit theorems for processes with infinite variance in the domain of attraction of stable laws.

Abstract

In this paper we study the genealogical structure of a Galton-Watson process with neutral mutations, where the initial population is large and mutation rate is small \cite{B2}. Namely, we extend in two directions the results obtained in Bertoin's work. In the critical case, we construct the version of Bertoin's model conditioned not to be extinct, and in the case with finite variance we show convergence after normalization, of allelic sub-populations towards a tree indexed CSBP with immigration. Besides, we establish the version of the limit theorems in \cite{B2}, been for the unconditioned process and for the process conditioned to non-extinction, in the case where the reproduction law has infinite variance and it is in the domain of attraction of an $\alpha$-stable distribution.