On the block counting process and the fixation line of exchangeable coalescents

TL;DR

This paper analyzes the block counting process and fixation line of exchangeable coalescents, providing formulas, duality relations, and convergence results, with detailed studies on Dirichlet and Poisson-Dirichlet coalescents.

Contribution

It introduces formulas for infinitesimal rates, establishes duality between processes, and proves convergence results for large sample sizes in exchangeable coalescents.

Findings

Block counting process is Siegmund dual to the fixation line.

Convergence of block counting process as sample size grows, related to singleton frequencies.

Detailed analysis of Dirichlet and Poisson-Dirichlet coalescents.

Abstract

We study the block counting process and the fixation line of exchangeable coalescents. Formulas for the infinitesimal rates of both processes are provided. It is shown that the block counting process is Siegmund dual to the fixation line. For exchangeable coalescents restricted to a sample of size n and with dust we provide a convergence result for the block counting process as n tends to infinity. The associated limiting process is related to the frequencies of singletons of the coalescent. Via duality we obtain an analog convergence result for the fixation line of exchangeable coalescents with dust. The Dirichlet coalescent and the Poisson-Dirichlet coalescent are studied in detail.