Asymptotic genealogies for a class of generalized Wright–Fisher models

Thierry Huillet,Martin Möhle

doi:10.15559/21-vmsta196

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.

Highlights

Let X1, X2, . . . be independent copies of a random variable X taking values in (0, ∞)
Consider the Cannings model [6, 7] with population size N and nonoverlapping generations such that, conditional on W1, . . . , WN, the offspring sizes ν1, . . . , νN have a multinomial distribution with parameters N and W1, . . . , WN
We introduce the effective population size Ne := 1/cN

Summary

Introduction

Let X1, X2, . . . be independent copies of a random variable X taking values in (0, ∞). (ii) Analogously, a Cannings model is said to be in the domain of attraction of a discrete-time coalescent = ( r )r∈N0 if for each sample size n ∈ N the ancestral process The proofs are provided in the main Section 4 They are based on general convergence-to-the-coalescent theorems for Cannings models provided in [32] and combine (Abelian and Tauberian) arguments from the theory of regularly varying functions in the spirit of Karamata [20,21,22] with techniques used by Huillet [15] for the Pareto case and by Schweinsberg [37] for the related model where the sampling is performed without replacement

Results

Examples

Proofs

A Appendix