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


Baur, Erich; Bertoin, Jean (2017). Weak limits for the largest subpopulations in Yule processes with high mutation probabilities. Advances in Applied Probability, 49(3):877-902.

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.

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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Additional indexing

Item Type:Journal Article, refereed, original work
Communities & Collections:07 Faculty of Science > Institute of Mathematics
Dewey Decimal Classification:510 Mathematics
Scopus Subject Areas:Physical Sciences > Statistics and Probability
Physical Sciences > Applied Mathematics
Language:English
Date:1 September 2017
Deposited On:31 Oct 2017 16:31
Last Modified:17 Jul 2024 01:38
Publisher:Cambridge University Press
ISSN:0001-8678
OA Status:Closed
Publisher DOI:https://doi.org/10.1017/apr.2017.25
Other Identification Number:Pre-Print Version auf arXiv.com: 10.48550/arXiv.1603.06564 (DOI)