The fixation line in the $\Lambda$-coalescent

Abstract

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 $n$-$\mathrm{Beta}(2-\alpha ,\alpha )$-coalescent is proved to converge as $n\to \infty$, and the generating function of the limiting random variable is computed.

Published at http://dx.doi.org/10.1214/14-AAP1077 in the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org)

References in corpus (2)

• Random recursive trees and the Bolthausen-Sznitman coalescent
• $\beta$-coalescents and stable Galton-Watson trees

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