On the size of the block of 1 for Ξ-coalescents with dust

Freund, Fabian; Möhle, Martin

doi:10.15559/17-VMSTA92

Modern Stochastics: Theory and Applications

On the size of the block of 1 for Ξ-coalescents with dust

Volume 4, Issue 4 (2017), pp. 407–425

Fabian Freund Martin Möhle

https://doi.org/10.15559/17-VMSTA92

Pub. online: 27 December 2017 Type: Research Article

Open Access

Received
28 August 2017

Revised
4 December 2017

Accepted
6 December 2017

Published
27 December 2017

Abstract

We study the frequency process $f_{1}$ of the block of 1 for a Ξ-coalescent Π with dust. If Π 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-Λ-coalescents with $\varLambda =\delta _{p}$, $p\in [\frac{1}{2},1)$, $f_{1}$ is not Markovian, whereas its jump chain is Markovian. For simple Λ-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.

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Keywords

Ξ-coalescent coalescent with dust Poisson point process minimal clade exchangeability

MSC2010

60F15 60J75 60G55 60G09

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