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

  • Erich Baur (a1) and Jean Bertoin (a2)
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.

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Corresponding author
* 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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Advances in Applied Probability
  • ISSN: 0001-8678
  • EISSN: 1475-6064
  • URL: /core/journals/advances-in-applied-probability
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