Asymptotic genealogies for a class of generalized Wright-Fisher models. (English) Zbl 1486.60115

Huillet, Thierry; Möhle, Martin

Mod. Stoch., Theory Appl. 9, No. 1, 17-43 (2022).

Summary: A class of Cannings models is studied, with population size \(N\) having a mixed multinomial offspring distribution with random success probabilities \({W_1},\dots ,{W_N}\) induced by independent and identically distributed positive random variables \({X_1},{X_2},\dots\) via \({W_i}:={X_i}/{S_N}\), \(i\in \{1,\dots ,N\}\), where \({S_N}:={X_1}+\cdots +{X_N}\). The ancestral lineages are hence based on a sampling with replacement strategy from a random partition of the unit interval into \(N\) subintervals of lengths \({W_1},\dots ,{W_N}\). Convergence results for the genealogy of these Cannings models are provided under assumptions that the tail distribution of \({X_1}\) is regularly varying. In the limit several coalescent processes with multiple and simultaneous multiple collisions occur. The results extend those obtained by T. E. Huillet [J. Math. Biol. 68, No. 3, 727–761 (2014; Zbl 1295.60083)] for the case when \({X_1}\) is Pareto distributed and complement those obtained by J. Schweinsberg [Stochastic Processes Appl. 106, No. 1, 107–139 (2003; Zbl 1075.60571)] for models where sampling is performed without replacement from a supercritical branching process.

MSC:

60J90	Coalescent processes
92D15	Problems related to evolution
60J80	Branching processes (Galton-Watson, birth-and-death, etc.)

Keywords:

Cannings model; exchangeable coalescent; regularly varying function; simultaneous multiple collisions; weak convergence

Citations:

Zbl 1295.60083; Zbl 1075.60571

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