Asymptotic genealogies for a class of generalized Wright–Fisher models

Abstract

We study a class of Cannings models with population size N having a mixed multinomial offspring distribution with random success probabilities W1, . . . ,WN induced by independent and identically distributed positive random variables X1, X2, . . . via Wi := Xi/SN , i ∈ {1, . . . , N}, where SN := X1 + · · · + XN . 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 W1, . . . ,WN . Convergence results for the genealogy of these Cannings models are provided under regularly varying assumptions on the tail distribution of X1. In the limit several coalescent processes with multiple and simultaneous multiple collisions occur. The results extend those obtained in [15] for the case when X1 is Pareto distributed and complement those obtained in [37] for models where one samples without replacement from a supercritical branching process.