• Corpus ID: 211096873

# Probabilistic aspects of $\Lambda$-coalescents in equilibrium and in evolution

@article{Kersting2020ProbabilisticAO,
title={Probabilistic aspects of \$\Lambda\$-coalescents in equilibrium and in evolution},
author={G{\"o}tz Kersting and A. Wakolbinger},
journal={arXiv: Probability},
year={2020}
}
• Published 12 February 2020
• Physics
• arXiv: Probability
We present approximation methods which lead to law of large numbers and fluctuation results for functionals of $\Lambda$-coalescents, both in the dust-free case and in the case with a dust component. Our focus is on the tree length (or total branch length) and the total external branch length, as well as the time to the most recent common ancestor and the size of the last merger. In the second part we discuss evolving coalescents. For certain Beta-coalescents we analyse fluctuations of a class…

## References

SHOWING 1-10 OF 48 REFERENCES
The size of the last merger and time reversal in $\Lambda$-coalescents
• Mathematics
Annales de l'Institut Henri Poincaré, Probabilités et Statistiques
• 2018
Author(s): Kersting, Goetz; Schweinsberg, Jason; Wakolbinger, Anton | Abstract: We consider the number of blocks involved in the last merger of a $\Lambda$-coalescent started with $n$ blocks. We give
Tree lengths for general $\Lambda$-coalescents and the asymptotic site frequency spectrum around the Bolthausen–Sznitman coalescent
• Mathematics
The Annals of Applied Probability
• 2019
We study tree lengths in $\Lambda$-coalescents without a dust component from a sample of $n$ individuals. For the total lengths of all branches and the total lengths of all external branches we
The asymptotic distribution of the length of Beta-coalescent trees
We derive the asymptotic distribution of the total length $L_n$ of a $\operatorname {Beta}(2-\alpha,\alpha)$-coalescent tree for $1<\alpha<2$, starting from $n$ individuals. There are two regimes: If
External branch lengths of Λ -coalescents without a dust component *
• Mathematics
• 2019
Λ -coalescents model genealogies of samples of individuals from a large population by means of a family tree. The tree’s leaves represent the individuals, and the lengths of the adjacent edges
The Λ-coalescent speed of coming down from infinity
• Mathematics
• 2010
Consider a $\Lambda$-coalescent that comes down from infinity (meaning that it starts from a configuration containing infinitely many blocks at time 0, yet it has a finite number $N_t$ of blocks at
The total external length of the evolving Kingman coalescent
• Mathematics
• 2014
The evolving Kingman coalescent is the tree-valued process which records the time evolution undergone by the genealogies of Moran populations. We consider the associated process of total external
The internal branch lengths of the Kingman coalescent
• Mathematics
• 2013
In the Kingman coalescent tree the length of order $r$ is defined as the sum of the lengths of all branches that support $r$ leaves. For $r=1$ these branches are external, while for $r\ge2$ they are
BETA-COALESCENTS AND CONTINUOUS STABLE RANDOM TREES
• Mathematics
• 2007
It is proved that Beta-coalescents can be embedded in continuous stable random trees based on a construction of the Donnelly-Kurtz lookdown process using continuous random trees, which is of independent interest.
The common ancestor type distribution of a $\Lambda$-Wright-Fisher process with selection and mutation
• Mathematics
• 2016
A strong pathwise Siegmund dual is identified of the ancestor in a two-type Wright-Fisher population with mutation and selection, conditional on the overall type frequency in the old population, and the equilibrium tail probabilities of $L$ are characterised in terms of hitting probabilities of the dual process.
Asymptotic sampling formulae for -coalescents
• Mathematics
• 2012
We present a robust method which translates information on the speed of coming down from infinity of a genealogical tree into sampling formulae for the underlying population. We apply these results