Abstract
We study a family of coalescent processes that undergo ``simultaneous multiple collisions,'' meaning that many clusters of particles can merge into a single cluster at one time, and many such mergers can occur simultaneously. This family of processes, which we obtain from simple assumptions about the rates of different types of mergers, essentially coincides with a family of processes that Mohle and Sagitov obtain as a limit of scaled ancestral processes in a population model with exchangeable family sizes. We characterize the possible merger rates in terms of a single measure, show how these coalescents can be constructed from a Poisson process, and discuss some basic properties of these processes. This work generalizes some work of Pitman, who provides similar analysis for a family of coalescent processes in which many clusters can coalesce into a single cluster, but almost surely no two such mergers occur simultaneously.
Highlights
We study a family of coalescent processes that undergo “simultaneous multiple collisions,” meaning that many clusters of particles can merge into a single cluster at one time, and many such mergers can occur simultaneously
These processes were previously introduced in [13] by Mohle and Sagitov, who obtained them by taking limits of scaled ancestral processes in a population model with exchangeable family sizes
We summarize the results needed to establish the two characterizations of the coalescents with simultaneous multiple collisions, one involving a single measure Ξ and the other involving a sequence of measures (Fr)∞ r=1
Summary
We study a family of coalescent processes that undergo “simultaneous multiple collisions,” meaning that many clusters of particles can merge into a single cluster at one time, and many such mergers can occur simultaneously. In [13], Mohle and Sagitov generalize the proofs in [12] and obtain coalescents with simultaneous multiple collisions as limits of ancestral processes in a haploid population model. Mohle and Sagitov leave open the question of whether every possible coalescent with simultaneous multiple collisions can be obtained as a limit of ancestral processes in their population model. They do not discuss the questions of which sequences of measures (Fr)∞ r=1 satisfying conditions A1, A2, and A3 of Proposition 1 are associated with coalescent processes in the manner described above, and whether there is a natural probabilistic interpretation of the measures Fr, aside from the interpretation of their moments as collision rates.
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
