A classification of coalescent processes for haploid exchangeable population models
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.
scientific article; zbMATH DE number 1906029