# The Total External Branch Length of Beta-Coalescents†

@article{Dahmer2014TheTE,
title={The Total External Branch Length of Beta-Coalescents†},
author={Iulia Dahmer and G{\"o}tz Kersting and A. Wakolbinger},
journal={Combinatorics, Probability and Computing},
year={2014},
volume={23},
pages={1010 - 1027}
}
• Published 25 December 2012
• Physics
• Combinatorics, Probability and Computing
For 1 < α < 2 we derive the asymptotic distribution of the total length of external branches of a Beta(2 − α, α)-coalescent as the number n of leaves becomes large. 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. This is in contrast to the fluctuation behaviour of the total branch length, which exhibits a transition at $\alpha_0 = (1… 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 J an 2 02 2 The joint fluctuations of the lengths of the Beta ( 2 − α , α )-coalescents ∗ • Mathematics • 2022 We consider Beta(2− α, α)-coalescents with parameter range 1 < α < 2 starting from n leaves. The length l (n) r of order r in the n-Beta(2 − α, α)-coalescent tree is defined as the sum of the lengths 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 whose branches have lengths. The tree’s leaves represent the individuals, and the lengths 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. A Note on the Small-Time Behaviour of the Largest Block Size of Beta n -Coalescents • Mathematics • 2018 We study the largest block size of Beta n-coalescents at small times as n tends to infinity, using the paintbox construction of Beta-coalescents and the link between continuous-state branching Asymptotics of the Minimal Clade Size and Related Functionals of Certain Beta-Coalescents • Mathematics • 2013 The Beta(2−α,α) n-coalescent with 1<α<2 is a Markov process taking values in the set of partitions of {1,…,n}. It evolves from the initial value {1},…,{n} by merging (coalescing) blocks together into 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 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$individuals. For the total lengths of all branches and the total lengths of all external branches we The joint fluctuations of the lengths of the Beta$(2-\alpha, \alpha)$-coalescents • Mathematics • 2020 We consider Beta$(2-\alpha, \alpha)-n$-coalescents with parameter range$1 <\alpha<2$. The length$\ell^{(n)}_r$of order$r$in the Beta$(2-\alpha, \alpha)-n$-coalescent tree is defined as the sum 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 ## References SHOWING 1-10 OF 32 REFERENCES ASYMPTOTIC RESULTS ABOUT THE TOTAL BRANCH LENGTH OF THE BOLTHAU-SEN-SZNITMAN COALESCENT • Mathematics • 2006 Abstract We study the total branch length Ln of the Bolthausen-Sznitman coalescent as the sample size n tends to infinity. Asymptotic expansions for the moments of Ln are presented. It is shown that 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 Small-time behavior of beta coalescents • Mathematics • 2008 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 On the Total External Length of the Kingman Coalescent • Mathematics • 2011 We prove asymptotic normality of the total length of external branches in the Kingman coalescent. The proof uses an embedded Markov chain, which can be described as follows: Take an urn with black Asymptotic results on the length of coalescent trees • Mathematics • 2007 We give the asymptotic distribution of the length of partial coalescent trees for Beta and related coalescents. This allows us to give the asymptotic distribution of the number of (neutral) mutations The general coalescent with asynchronous mergers of ancestral lines 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 Asympotic behavior of the total length of external branches for Beta-coalescents • Mathematics • 2012 It is proved that$n^{{\alpha}-2}L^{(n)_{ext}$converges in$L^2$to$\alpha(\alpha-1)\Gamma(\alpha)$as a consequence of the asymptotic behavior of the total length of the associated$n\$-coalescent.
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.