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Weak limits for the largest subpopulations in Yule processes with high mutation probabilities

Part of: Special processes Markov processes

Published online by Cambridge University Press:  08 September 2017

Erich Baur  and
Jean Bertoin
Erich Baur*
Affiliation:
ENS de Lyon
Jean Bertoin*
Affiliation:
Universität Zürich
*
* Current address: Bern University Of Applied Sciences, Quellgasse 21, 2501 Biel, Switzerland. Email address: erich.baur@bfh.ch
** Postal address: Institut für Mathematik, Universität Zürich, Winterthurerstrasse 190, CH-8057 Zürich, Switzerland. Email address: jean.bertoin@math.uzh.ch
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Abstract

We consider a Yule process until the total population reaches size n ≫ 1, and assume that neutral mutations occur with high probability 1 - p (in the sense that each child is a new mutant with probability 1 - p, independently of the other children), where p = p n ≪ 1. We establish a general strategy for obtaining Poisson limit laws and a weak law of large numbers for the number of subpopulations exceeding a given size and apply this to some mutation regimes of particular interest. Finally, we give an application to subcritical Bernoulli bond percolation on random recursive trees with percolation parameter p n tending to 0.

Keywords

Branching processmutationpercolationrandom increasing tree

MSC classification

Primary: 60J27: Continuous-time Markov processes on discrete state spaces
Secondary: 60J80: Branching processes (Galton-Watson, birth-and-death, etc.) 60K35: Interacting random processes; statistical mechanics type models; percolation theory

Information

Type
Research Article
Information
Advances in Applied Probability , Volume 49 , Issue 3 , September 2017 , pp. 877 - 902
Copyright
Copyright © Applied Probability Trust 2017 

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