The symmetric coalescent and Wright–Fisher models with bottlenecks

Abstract

We define a new class of Ξ-coalescents characterized by a possibly infinite measure over the nonnegative integers. We call them symmetric coalescents since they are the unique family of exchangeable coalescents satisfying a symmetry property on their coagulation rates: they are invariant under any transformation that consists of moving one element from one block to another without changing the total number of blocks. We illustrate the diversity of behaviors of this family of processes by introducing and studying a one parameter subclass, the $(\mathit{\beta },\mathit{S})$-coalescents. We also embed this family in a larger class of Ξ-coalescents arising as the limit genealogies of Wright–Fisher models with bottlenecks. Some convergence results rely on a new Skorokhod type metric, that induces the Meyer–Zheng topology, which allows us to study the scaling limit of non-Markovian processes using standard techniques.