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2018 Coalescent results for diploid exchangeable population models
Matthias Birkner, Huili Liu, Anja Sturm
Electron. J. Probab. 23: 1-44 (2018). DOI: 10.1214/18-EJP175

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.

Citation

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Matthias Birkner. Huili Liu. Anja Sturm. "Coalescent results for diploid exchangeable population models." Electron. J. Probab. 23 1 - 44, 2018. https://doi.org/10.1214/18-EJP175

Information

Received: 9 September 2017; Accepted: 30 April 2018; Published: 2018
First available in Project Euclid: 1 June 2018

zbMATH: 06924661
MathSciNet: MR3814243
Digital Object Identifier: 10.1214/18-EJP175

Subjects:
Primary: 60J17, 92D10
Secondary: 60J70, 92D25

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