Minimal Clade Size in the Bolthausen-Sznitman Coalescent

@article{Freund2014MinimalCS,
  title={Minimal Clade Size in the Bolthausen-Sznitman Coalescent},
  author={Fabian Freund and Arno Siri-J{\'e}gousse},
  journal={Journal of Applied Probability},
  year={2014},
  volume={51},
  pages={657 - 668}
}
In this article we show the asymptotics of distribution and moments of the size X n of the minimal clade of a randomly chosen individual in a Bolthausen-Sznitman n-coalescent for n → ∞. The Bolthausen-Sznitman n-coalescent is a Markov process taking states in the set of partitions of {1, …, n}, where 1, …, n are referred to as individuals. The minimal clade of an individual is the equivalence class the individual is in at the time of the first coalescence event this individual participates in… 
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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 β (3 / 2 , 1 /
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Journal of Applied Probability Volume 51 (2014): Index
  • Mathematics
    Journal of Applied Probability
  • 2014
pages Albrecher, H., Boxma, O. J. and Ivanovs, J. On simple ruin expressions in dependent Sparre Andersen risk models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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References

SHOWING 1-10 OF 23 REFERENCES
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
Coalescents with multiple collisions
k−2 � 1 − xb−k � � dx� . Call this process a � -coalescent. Discrete measure-valued processes derived from the � -coalescent model a system of masses undergoing coalescent collisions. Kingman's
Minimal clade size and external branch length under the neutral coalescent
Given a sample of genes taken from a large population, we consider the neutral coalescent genealogy and study the theoretical and empirical distributions of the size of the smallest clade containing
On Asymptotics of Exchangeable Coalescents with Multiple Collisions
We study the number of collisions, X n , of an exchangeable coalescent with multiple collisions (Λ-coalescent) which starts with n particles and is driven by rates determined by a finite
Genealogies in simple models of evolution
TLDR
In the asexual case, selection leads to coalescence times which grow logarithmically with the size of the population, in contrast with the linear growth of the neutral case.
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
THE COALESCENT
The genealogy of branching Brownian motion with absorption
We consider a system of particles which perform branching Brownian motion with negative drift and are killed upon reaching zero, in the near-critical regime where the total population stays roughly
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