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

Abstract

We study the frequency process $f_1$ of the block of 1 for a $\Xi$-coalescent $\Pi$ with dust. If $\Pi$ 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-$\Lambda$-coalescents with $\Lambda ={\delta }_p$, $p\in [\frac{1}{2},1)$, $f_1$ is not Markovian, whereas its jump chain is Markovian. For simple $\Lambda$-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.

Published at https://doi.org/10.15559/17-VMSTA92 in the Modern Stochastics: Theory and Applications (https://www.i-journals.org/vtxpp/VMSTA) by VTeX (http://www.vtex.lt/)

References in corpus (2)

• Random recursive trees and the Bolthausen-Sznitman coalescent
• The size of the last merger and time reversal in $\Lambda$-coalescents