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.
%0 Journal Article
%1 schweinsberg2003coalescent
%A Schweinsberg, Jason
%D 2003
%J Stochastic Processes and their Applications
%K branching_processes coalescent_theory lambda_coalescent
%N 1
%P 107-139
%R https://doi.org/10.1016/S0304-4149(03)00028-0
%T Coalescent processes obtained from supercritical Galton--Watson processes
%U https://www.sciencedirect.com/science/article/pii/S0304414903000280
%V 106
%X 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.
@article{schweinsberg2003coalescent,
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.},
added-at = {2021-06-29T18:04:45.000+0200},
author = {Schweinsberg, Jason},
biburl = {https://www.bibsonomy.org/bibtex/2b3c8b4c8d39061583c77198a2eb1cd00/peter.ralph},
doi = {https://doi.org/10.1016/S0304-4149(03)00028-0},
interhash = {b6d6d230489e2b47d4d1398f8cbb6305},
intrahash = {b3c8b4c8d39061583c77198a2eb1cd00},
issn = {0304-4149},
journal = {Stochastic Processes and their Applications},
keywords = {branching_processes coalescent_theory lambda_coalescent},
number = 1,
pages = {107-139},
timestamp = {2021-06-29T18:04:45.000+0200},
title = {Coalescent processes obtained from supercritical {Galton}--{Watson} processes},
url = {https://www.sciencedirect.com/science/article/pii/S0304414903000280},
volume = 106,
year = 2003
}