Coalescent results for diploid exchangeable population models. (English) Zbl 1415.92122

Summary: 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 \(V=(V_{i,j})_{1\leq i \neq j \leq N}\) 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. Möhle and S. Sagitov’s [Ann. Probab. 29, No. 4, 1547–1562 (2001; Zbl 1013.92029)] result for the haploid case and sharpens the profile of M. Möhle and S. Sagitov [J. Math. Biol. 47, No. 4, 337–352 (2003; Zbl 1054.92039)] 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 J. Schweinsberg’s model [Stochastic Processes Appl. 106, No. 1, 107–139 (2003; Zbl 1075.60571)], leading to Beta-coalescents with two-fold and with four-fold mergers, respectively.


92D10 Genetics and epigenetics
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