## 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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### References:

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