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}
}
  • Olivier H'enard
  • Published 2 July 2013
  • Mathematics
  • Annals of Applied Probability
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… 

Figures from this paper

$\beta$-coalescents and stable Galton-Watson trees
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
On the size of the block of 1 for $\varXi$-coalescents with dust
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
On the block counting process and the fixation line of exchangeable coalescents
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
The sequential loss of allelic diversity Submitted to appear in the Festschrift in honor of Peter Jagers
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
The sequential loss of allelic diversity
TLDR
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.
The rate of convergence of the block counting process of exchangeable coalescents with dust
  • M. Möhle
  • Mathematics
    Latin American Journal of Probability and Mathematical Statistics
  • 2021
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
...
...

References

SHOWING 1-10 OF 29 REFERENCES
Alpha-Stable Branching and Beta-Coalescents
We determine that the continuous-state branching processes for which the genealogy, suitably time-changed, can be described by an autonomous Markov process are precisely those arising from
$\beta$-coalescents and stable Galton-Watson trees
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
Random Recursive Trees and the Bolthausen-Sznitman Coalesent
We describe a representation of the Bolthausen-Sznitman coalescent in terms of the cutting of random recursive trees. Using this representation, we prove results concerning the final collision of the
The general coalescent with asynchronous mergers of ancestral lines
  • S. Sagitov
  • Mathematics
    Journal of Applied Probability
  • 1999
Take a sample of individuals in the fixed-size population model with exchangeable family sizes. Follow the ancestral lines for the sampled individuals backwards in time to observe the ancestral
On hitting probabilities of beta coalescents and absorption times of coalescents that come down from infinity
Let X = (Xk)k=0;1;::: denote the jump chain of the block counting process of the -coalescent with = (2 ; ) being the beta distribution with parameter 2 (0;2). A solution for the hitting probability
Stochastic flows associated to coalescent processes
Abstract. We study a class of stochastic flows connected to the coalescent processes that have been studied recently by Möhle, Pitman, Sagitov and Schweinsberg in connection with asymptotic models
On Asymptotics of the Beta Coalescents
We show that the total number of collisions in the exchangeable coalescent process driven by the beta (1, b) measure converges in distribution to a 1-stable law, as the initial number of particles
-coalescents and stable Galton-Watson trees
. 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 β (3 / 2 , 1 /
Small-time behavior of beta coalescents
For a finite measureon (0,1), the �-coalescent is a coalescent process such that, whenever there are b clusters, each k-tuple of clusters merges into one at rate R 1 0 x k 2 (1 x) b k �(dx). It has
...
...