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A classification of coalescent processes for haploid exchangeable population models. (English) Zbl 1013.92029

This paper is concerned with haploid population models in which the population size \(N\) is fixed, and the generations are non-overlapping. For any specific \(r\)-th generation, it is assumed that the family sizes \[ \nu_1^{(r)},\dots,\nu_i^{(r)}, \dots,\nu_N^{(r)}, \] of the offspring of the individuals \(i = 1,\dots,N\), alive in this generation, are exchangeable random variables. After appropriate scaling of the ancestral process, the authors establish a weak convergence criterion for these family sizes as \(N\to \infty\). This allows a full classification of the coalescent generators for exchangeable reproduction.
Apart from the statement of the relevant theorem and its lengthy proof, the paper devotes a section to the particular population model whose time-scaled ancestral process converges to that of the Wright-Fisher model. The authors conclude with a discussion of their results, and provide a useful historical perspective of coalescent processes.

MSC:

92D15 Problems related to evolution
60J70 Applications of Brownian motions and diffusion theory (population genetics, absorption problems, etc.)
60F17 Functional limit theorems; invariance principles
92D25 Population dynamics (general)
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