Asymptotic genealogies for a class of generalized Wright–Fisher models
Open Access
- 15 December 2021
- journal article
- research article
- Published by VTeX in Modern Stochastics: Theory and Applications
- Vol. 9 (1), 17-43
- https://doi.org/10.15559/21-vmsta196
Abstract
Publisher: VTeX - Solutions for Science Publishing, Journal: Modern Stochastics - Theory and Applications, Title: Asymptotic genealogies for a class of generalized Wright–Fisher models, Authors: Thierry Huillet, Martin Möhle , A class of Cannings models is studied, with population size N having a mixed multinomial offspring distribution with random success probabilities ${W_{1}},\dots ,{W_{N}}$ induced by independent and identically distributed positive random variables ${X_{1}},{X_{2}},\dots $ via ${W_{i}}:={X_{i}}/{S_{N}}$, $i\in \{1,\dots ,N\}$, where ${S_{N}}:={X_{1}}+\cdots +{X_{N}}$. The ancestral lineages are hence based on a sampling with replacement strategy from a random partition of the unit interval into N subintervals of lengths ${W_{1}},\dots ,{W_{N}}$. Convergence results for the genealogy of these Cannings models are provided under assumptions that the tail distribution of ${X_{1}}$ is regularly varying. In the limit several coalescent processes with multiple and simultaneous multiple collisions occur. The results extend those obtained by Huillet [J. Math. Biol. 68 (2014), 727–761] for the case when ${X_{1}}$ is Pareto distributed and complement those obtained by Schweinsberg [Stoch. Process. Appl. 106 (2003), 107–139] for models where sampling is performed without replacement from a supercritical branching process.
Keywords
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