Coalescent processes obtained from supercritical Galton-Watson processes

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 --> infinity 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 greater than or equal to k) similar to 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. (C) 2003 Elsevier Science B.V. All rights reserved.