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

Abstract : 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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Journal articles
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https://hal.archives-ouvertes.fr/hal-00805322
Contributor : Romain Abraham Connect in order to contact the contributor
Submitted on : Wednesday, January 7, 2015 - 4:05:27 PM
Last modification on : Tuesday, October 12, 2021 - 5:20:12 PM
Long-term archiving on: : Wednesday, April 8, 2015 - 12:25:40 PM

### Files

levy-coalescent-soumis.pdf
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### Identifiers

• HAL Id : hal-00805322, version 2
• ARXIV : 1303.6882

### Citation

Romain Abraham, Jean-François Delmas. $\beta$-coalescents and stable Galton-Watson trees. ALEA : Latin American Journal of Probability and Mathematical Statistics, Instituto Nacional de Matemática Pura e Aplicada, 2015, 12, pp.451-476. ⟨hal-00805322v2⟩

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