# The fixation line in the ${\Lambda}$-coalescent

@article{Henard2015TheFL, title={The fixation line in the \$\{\Lambda\}\$-coalescent}, author={Olivier H'enard}, journal={Annals of Applied Probability}, year={2015}, volume={25}, pages={3007-3032} }

We define a Markov process in a forward population model with backward genealogy given by the $\Lambda$-coalescent. This Markov process, called the fixation line, is related to the block counting process through its hitting times. Two applications are discussed. The probability that the $n$-coalescent is deeper than the $(n-1)$-coalescent is studied. The distribution of the number of blocks in the last coalescence of the $n$-$\operatorname {Beta}(2-\alpha,\alpha)$-coalescent is proved to…

## 13 Citations

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

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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…

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We study the block counting process and the fixation line of exchangeable coalescents. Formulas for the infinitesimal rates of both processes are provided. It is shown that the block counting process…

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This paper gives a new flavor of what Peter Jagers and his co-authors call ‘the path to extinction’. In a neutral population with constant size N , we assume that each individual at time 0 carries a…

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It is shown that in the Moran model, the extinction process is distributed as the process counting the number of common ancestors to the whole population, also known as the block counting process of the N-Kingman coalescent, which extends to the general case of Λ-Fleming‒Viot processes.

On the block counting process and the fixation line of the Bolthausen-Sznitman coalescent

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The rate of convergence of the block counting process
of exchangeable coalescents with dust

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For exchangeable coalescents with dust the rate of convergence as the sample size tends to infinity of the scaled block counting process to the frequency of singleton process is determined. This rate…

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