On branching process with rare neutral mutation

Abstract

In this paper, we study the genealogical structure of a Galton–Watson process with neutral mutations. Namely, we extend in two directions the asymptotic results obtained in Bertoin [Stochastic Process. Appl. 120 (2010) 678–697]. In the critical case, we construct the version of the model in Bertoin [Stochastic Process. Appl. 120 (2010) 678–697], conditioned not to be extinct. We establish a version of the limit theorems in Bertoin [Stochastic Process. Appl. 120 (2010) 678–697], when the reproduction law has an infinite variance and it is in the domain of attraction of an $\alpha$-stable distribution, both for the unconditioned process and for the process conditioned to nonextinction. In the latter case, we obtain the convergence (after re-normalization) of the allelic sub-populations towards a tree indexed CSBP with immigration.