Bernoulli

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

On branching process with rare neutral mutation

Airam Blancas and Víctor Rivero

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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


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