• Corpus ID: 16792134

# Asympotic behavior of the total length of external branches for Beta-coalescents

@article{Dhersin2012AsympoticBO,
title={Asympotic behavior of the total length of external branches for Beta-coalescents},
author={J. S. Dhersin and Linglong Yuan},
journal={arXiv: Probability},
year={2012}
}
• Published 26 February 2012
• Mathematics
• arXiv: Probability
We consider a ${\Lambda}$-coalescent and we study the asymptotic behavior of the total length $L^{(n)}_{ext}$ of the external branches of the associated $n$-coalescent. For Kingman coalescent, i.e. ${\Lambda}={\delta}_0$, the result is well known and is useful, together with the total length $L^{(n)}$, for Fu and Li's test of neutrality of mutations% under the infinite sites model asumption . For a large family of measures ${\Lambda}$, including Beta$(2-{\alpha},{\alpha})$ with $0<\alpha<1$, M…
7 Citations

## Figures from this paper

### 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

### Pareto genealogies arising from a Poisson branching evolution model with selection

• T. Huillet
• Mathematics
Journal of mathematical biology
• 2014
It is indicated that this class of coalescent processes (and their scaling limits) may be viewed as the genealogical processes of some forward in time evolving branching population models including selection effects.

### 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 Total External Branch Length of Beta-Coalescents†

• Physics
Combinatorics, Probability and Computing
• 2014
It turns out that the fluctuations of the external branch length follow those of τn2−α over the entire parameter regime, where τn denotes the random number of coalescences that bring the n lineages down to one.

### On the total length of external branches for beta-coalescents

• Mathematics
Advances in Applied Probability
• 2015
In this paper we consider the beta(2 − α, α)-coalescents with 1 < α < 2 and study the moments of external branches, in particular, the total external branch length of an initial sample of n

### On the external branches of coalescents with multiple collisions

• Mathematics, Physics
• 2012
A recursion for the joint moments of the external branch lengths for coalescents with multiple collisions (Lambda-coalescents) is provided and results show that the lengths of two randomly chosen external branches are positively correlated for the Bolthausen-Sznitman coalescent.

### Comportement en temps petits des Λ n-coalescents avec l'accent sur les longueurs des branches externes

Le Λ-coalescent est un processus stochastique utilise pour modeliser l'arbre genealogique d'une population composee d'une infinite d'individus, admettant des collisions multiples de lignages. Motivee

## References

SHOWING 1-10 OF 46 REFERENCES

### 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

### Alpha-Stable Branching and Beta-Coalescents

• Mathematics
• 2005
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

### A limiting distribution for the number of cuts needed to isolate the root of a random recursive tree

• Mathematics, Computer Science
Random Struct. Algorithms
• 2009
A weak convergence result is provided for Xn, the number of cuts needed to isolate the root in a random recursive tree with n vertices, which was not studied previously in the sense of weak convergence.

### Random Recursive Trees and the Bolthausen-Sznitman Coalesent

• Mathematics
• 2005
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 Total External Branch Length of Beta-Coalescents†

• Physics
Combinatorics, Probability and Computing
• 2014
It turns out that the fluctuations of the external branch length follow those of τn2−α over the entire parameter regime, where τn denotes the random number of coalescences that bring the n lineages down to one.

### Asymptotic sampling formulae and particle system representations for $\Lambda$-coalescents

• Mathematics
• 2011
Consider an evolving population, with genealogy given by a �-coalescent that comes down from infinity. We provide rather explicit sampling formulae under this model, for large samples. More