Coalescent results for diploid exchangeable population models

TL;DR

This paper establishes conditions under which the ancestral process of a diploid exchangeable population model converges to a oalescent, extending previous haploid results and applying to models with multiple mergers.

Contribution

It generalizes coalescent convergence results to diploid exchangeable models, including new examples with Beta-coalescents and multiple mergers.

Findings

Derived general convergence conditions for diploid models.

Extended previous haploid coalescent results to diploid case.

Applied results to models leading to Beta-coalescents with multiple mergers.

Abstract

We consider diploid bi-parental analogues of Cannings models: in a population of fixed size $N$ the next generation is composed of $V_{i,j}$ offspring from parents $i$ and $j$, where $V=(V_{i,j})_{1\le i\neq j \le N}$ is a (jointly) exchangeable (symmetric) array. Every individual carries two chromosome copies, each of which is inherited from one of its parents. We obtain general conditions, formulated in terms of the vector of the total number of offspring to each individual, for the convergence of the properly scaled ancestral process for an $n$-sample of genes towards a ($\Xi$-)coalescent. This complements M\"ohle and Sagitov's (2001) result for the haploid case and sharpens the profile of M\"ohle and Sagitov's (2003) study of the diploid case, which focused on fixed couples, where each row of $V$ has at most one non-zero entry. We apply the convergence result to several examples, in particular to two diploid variations of Schweinsberg's (2003) model, leading to Beta-coalescents with two-fold and with four-fold mergers, respectively.