Genealogies of regular exchangeable coalescents with applications to sampling

TL;DR

This paper analyzes the genealogy of regular exchangeable coalescents with mutations, deriving asymptotic results for lineages and implications for sampling theory, especially in models where genealogies come down from infinity.

Contribution

It provides new asymptotic formulas for active and mutation-free lineages, extending previous work with martingale techniques and linking genealogical properties to sampling models.

Findings

Asymptotic expressions for the number of active lineages.

Results on the number of mutation-free lineages.

Implications for sampling in Xi-coalescent models.

Abstract

This article considers a model of genealogy corresponding to a regular exchangeable coalescent (also known as Xi-coalescent) started from a large finite configuration, and undergoing neutral mutations. Asymptotic expressions for the number of active lineages were obtained by the author in a previous work. Analogous results for the number of active mutation-free lineages and the combined lineage lengths are derived using the same martingale-based technique. They are given in terms of convergence in probability, while extensions to convergence in moments and convergence almost surely are discussed. The above mentioned results have direct consequences on the sampling theory in the Xi-coalescent setting. In particular, the regular Xi-coalescents that come down from infinity (i.e., with locally finite genealogies), have an asymptotically equal number of families under the corresponding infinite alleles and infinite sites models. In special cases, quantitative asymptotic formulae for the number of families that contain a fixed number of individuals can be given.