On the size of the block of 1 for $\varXi$-coalescents with dust

TL;DR

This paper analyzes the size of the block containing element 1 in $ ext{Xi}$-coalescents with dust, revealing its jump-hold structure and distributional properties, especially for Dirac-$ ext{Lambda}$-coalescents.

Contribution

It provides a detailed description of the frequency process of block 1 in $ ext{Xi}$-coalescents with dust, including explicit distributional results and Markovian properties for specific cases.

Findings

The frequency process $f_1$ is a jump-hold process expressed via a stick-breaking procedure.

For Dirac-$ ext{Lambda}$-coalescents with $p o 1$, $f_1$ is not Markovian, but its jump chain is.

The distribution of $f_1$ at its first jump is given by shifted geometric distributions.

Abstract

We study the frequency process $f_1$ of the block of 1 for a $\varXi$-coalescent $\varPi$ with dust. If $\varPi$ stays infinite, $f_1$ is a jump-hold process which can be expressed as a sum of broken parts from a stick-breaking procedure with uncorrelated, but in general non-independent, stick lengths with common mean. For Dirac-$\varLambda$-coalescents with $\varLambda=\delta_p$, $p\in[\frac{1}{2},1)$, $f_1$ is not Markovian, whereas its jump chain is Markovian. For simple $\varLambda$-coalescents the distribution of $f_1$ at its first jump, the asymptotic frequency of the minimal clade of 1, is expressed via conditionally independent shifted geometric distributions.