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Asymptotic genealogies for a class of generalized Wright–Fisher models
Volume 9, Issue 1 (2022), pp. 17–43
Thierry Huillet   Martin Möhle ORCID icon link to view author Martin Möhle details  

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https://doi.org/10.15559/21-VMSTA196
Pub. online: 15 December 2021      Type: Research Article      Open accessOpen Access

Received
8 March 2021
Revised
30 September 2021
Accepted
27 November 2021
Published
15 December 2021

Abstract

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.

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Keywords
Cannings model exchangeable coalescent regularly varying function simultaneous multiple collisions weak convergence 60J90 92D15

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