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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: [email protected]
** Postal address: Institut für Mathematik, Universität Zürich, Winterthurerstrasse 190, CH-8057 Zürich, Switzerland. Email address: [email protected]
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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 = pn ≪ 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 pn 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
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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