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Pré-Publication, Document De Travail Année : 2013

$\beta$-coalescents and stable Galton-Watson trees

Résumé

Representation of coalescent process using pruning of trees has been used by Goldschmidt and Martin for the Bolthausen-Sznitman coalescent and by Abraham and Delmas for the $\beta(3/2,1/2)$-coalescent. By considering a pruning procedure on stable Galton-Watson tree with $n$ labeled leaves, we give a representation of the discrete $\beta(1+\alpha,1-\alpha)$-coalescent, with $\alpha\in [1/2,1)$ starting from the trivial partition of the $n$ first integers. The construction can also be made directly on the stable continuum Lévy tree, with parameter $1/\alpha$, simultaneously for all $n$. This representation allows to use results on the asymptotic number of coalescence events to get the asymptotic number of cuts in stable Galton-Watson tree (with infinite variance for the reproduction law) needed to isolate the root. Using convergence of the stable Galton-Watson tree conditioned to have infinitely many leaves, one can get the asymptotic distribution of blocks in the last coalescence event in the $\beta(1+\alpha,1-\alpha)$-coalescent.
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Dates et versions

hal-00805322 , version 1 (27-03-2013)
hal-00805322 , version 2 (07-01-2015)

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Romain Abraham, Jean-François Delmas. $\beta$-coalescents and stable Galton-Watson trees. 2013. ⟨hal-00805322v1⟩
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