Article,
Coalescent processes obtained from supercritical Galton--Watson processes
Stochastic Processes and their Applications, 106 (1): 107-139 (2003)
DOI: https://doi.org/10.1016/S0304-4149(03)00028-0
Abstract
Consider a population model in which there are N individuals in each generation. One can obtain a coalescent tree by sampling n individuals from the current generation and following their ancestral lines backwards in time. It is well-known that under certain conditions on the joint distribution of the family sizes, one gets a limiting coalescent process as N→∞ after a suitable rescaling. Here we consider a model in which the numbers of offspring for the individuals are independent, but in each generation only N of the offspring are chosen at random for survival. We assume further that if X is the number of offspring of an individual, then P(X⩾k)∼Ck−a for some a>0 and C>0. We show that, depending on the value of a, the limit may be Kingman's coalescent, in which each pair of ancestral lines merges at rate one, a coalescent with multiple collisions, or a coalescent with simultaneous multiple collisions.
