Asymptotic genealogies for a class of generalized Wright-Fisher models

TL;DR

This paper analyzes the ancestral genealogies of a class of Cannings models with random offspring distributions, revealing convergence to complex coalescent processes under specific tail assumptions.

Contribution

It extends previous work by characterizing the asymptotic genealogies of generalized Wright-Fisher models with heavy-tailed offspring distributions.

Findings

Convergence to multiple and simultaneous multiple collision coalescents.

Extension of Huillet (2014) results for Pareto distributions.

Complementary to Schweinsberg (2003) models with sampling without replacement.

Abstract

We study a class of Cannings models with population size $N$ having a mixed multinomial offspring distribution with random success probabilities $W_1,\ldots,W_N$ induced by independent and identically distributed positive random variables $X_1,X_2,\ldots$ via $W_i:=X_i/S_N$, $i\in\{1,\ldots,N\}$, 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 $W_1,\ldots,W_N$. Convergence results for the genealogy of these Cannings models are provided under regularly varying assumptions on the tail distribution of $X_1$. In the limit several coalescent processes with multiple and simultaneous multiple collisions occur. The results extend those obtained by Huillet (2014) for the case when $X_1$ is Pareto distributed and complement those obtained by Schweinsberg (2003) for models where one samples without replacement from a supercritical branching process.