| 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 Möhle 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. We also provide some results on the problem of determining under what conditions the number of clusters is finite for all positive times, even if the number of clusters is infinite at time zero. This work generalizes some work of Pitman, who provides similar analysis for “coalescents with multiple collisions,” a family of coalescent processes in which many clusters can coalesce into a single cluster, but almost surely no two such mergers occur simultaneously. |
