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 $W_1,\dots ,W_N$ 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 regularly varying assumptions on the tail distribution of $X_1$. In the limit several coalescent processes with multiple and simultaneous multiple collisions occur. The results extend those obtained by Huillet (2014) for the case when $X_1$ is Pareto distributed and complement those obtained by Schweinsberg (2003) for models where one samples without replacement from a supercritical branching process.

24 pages