## Bernoulli

- Bernoulli
- Volume 24, Number 2 (2018), 1576-1612.

### On branching process with rare neutral mutation

Airam Blancas and Víctor Rivero

#### 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.

#### Article information

**Source**

Bernoulli, Volume 24, Number 2 (2018), 1576-1612.

**Dates**

Received: August 2015

Revised: June 2016

First available in Project Euclid: 21 September 2017

**Permanent link to this document**

https://projecteuclid.org/euclid.bj/1505980904

**Digital Object Identifier**

doi:10.3150/16-BEJ907

**Mathematical Reviews number (MathSciNet)**

MR3706802

**Zentralblatt MATH identifier**

06778373

**Keywords**

branching process domain of attraction of $\alpha$-stable laws neutral mutations Q-processes regular variation

#### Citation

Blancas, Airam; Rivero, Víctor. On branching process with rare neutral mutation. Bernoulli 24 (2018), no. 2, 1576--1612. doi:10.3150/16-BEJ907. https://projecteuclid.org/euclid.bj/1505980904