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

Fabian Freund, Martin Möhle

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.

Keywords


$\varXi$-coalescent, coalescent with dust, Poisson point process, minimal clade, exchangeability

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References


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DOI: http://dx.doi.org/10.15559/17-VMSTA92

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