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**Robustness results for the coalescent.**
*(English)*
Zbl 0913.60022

This paper is concerned with a convergence theorem to the coalescent in a genealogical process. The usual conditions required for such theorems to hold rely on assumptions of exchangeability; the author dispenses with these in his paper. By an analysis of the coalescence probability, i.e., the probability that two genes chosen at random without replacement from a gene population of size \(N\) have a common ancestor in the previous generation, the author is able to prove his main theorem. This states that for a large class of non-exchangeable population models, there is convergence of the finite-dimensional distributions of a certain well defined process \(\{R_{\tau_N}(t)\}_{t\in T}\) to those of the \(n\)-coalescent, as the number \(N\) of genes in the population tends to infinity. This generalizes the earlier proof of J. F. C. Kingman [in: Exchangeability in probability and statistics, 97-112 (1982; Zbl 0494.92011)] for a class of exchangeable models.

Reviewer: J.M.Gani (Canberra)

### MSC:

60F05 | Central limit and other weak theorems |

92D10 | Genetics and epigenetics |

60J27 | Continuous-time Markov processes on discrete state spaces |

92D25 | Population dynamics (general) |