Electronic Journal of Probability

Coalescent results for diploid exchangeable population models  

Matthias Birkner, Huili Liu, and Anja Sturm

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

Permanent link to this document
https://projecteuclid.org/euclid.ejp/1527818427

Digital Object Identifier
doi:10.1214/18-EJP175

Zentralblatt MATH identifier
06924661

Subjects
Primary: 60J17 92D10: Genetics {For genetic algebras, see 17D92}
Secondary: 60J70: Applications of Brownian motions and diffusion theory (population genetics, absorption problems, etc.) [See also 92Dxx] 92D25: Population dynamics (general)

Keywords
diploid population model diploid ancestral process coalescent with simultaneous multiple collisions

Rights
Creative Commons Attribution 4.0 International License.

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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