# 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} }

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

### 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

- MathematicsJournal 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†

- PhysicsCombinatorics, 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

- MathematicsAdvances 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

- 2013

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…

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