## Electronic Journal of Probability

### Coalescent results for diploid exchangeable population models

#### 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 Möhle and Sagitov’s (2001) result for the haploid case and sharpens the profile of Möhle 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.

#### Article information

Source
Electron. J. Probab., Volume 23 (2018), paper no. 49, 44 pp.

Dates
Accepted: 30 April 2018
First available in Project Euclid: 1 June 2018

https://projecteuclid.org/euclid.ejp/1527818427

Digital Object Identifier
doi:10.1214/18-EJP175

Zentralblatt MATH identifier
06924661

#### Citation

Birkner, Matthias; Liu, Huili; Sturm, Anja. Coalescent results for diploid exchangeable population models. Electron. J. Probab. 23 (2018), paper no. 49, 44 pp. doi:10.1214/18-EJP175. https://projecteuclid.org/euclid.ejp/1527818427

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