# Asymptotic results concerning the total branch length of the Bolthausen-Sznitman coalescent

@article{Drmota2007AsymptoticRC, title={Asymptotic results concerning the total branch length of the Bolthausen-Sznitman coalescent}, author={Michael Drmota and Alexander Iksanov and Martin Moehle and Uwe Roesler}, journal={Stochastic Processes and their Applications}, year={2007}, volume={117}, pages={1404-1421} }

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## 55 Citations

### ON THE EXTERNAL BRANCHES OF COALESCENT PROCESSES WITH MULTIPLE COLLISIONS WITH AN EMPHASIS ON THE BOLTHAUSEN-SZNITMAN COA- LESCENT

- Mathematics, Physics
- 2012

A recursion for the joint moments of the external branch lengths for coalescents with multiple collisions (-coalescents) is provided. This recursion is used to derive asymptotic expansions as the…

### On Asymptotics of the Beta Coalescents

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We show that the total number of collisions in the exchangeable coalescent process driven by the beta (1, b) measure converges in distribution to a 1-stable law, as the initial number of particles…

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

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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.

### Tree lengths for general $\Lambda $-coalescents and the asymptotic site frequency spectrum around the Bolthausen–Sznitman coalescent

- MathematicsThe 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…

### Absorption Time and Tree Length of the Kingman Coalescent and the Gumbel Distribution

- Mathematics
- 2015

Formulas are provided for the cumulants and the moments of the time T back to the most recent common ancestor of the Kingman coalescent. It is shown that both the jth cumulant and the jth moment of T…

### Asymptotic hitting probabilities for the Bolthausen-Sznitman coalescent

- MathematicsJournal of Applied Probability
- 2014

The probability h(n, m) that the block counting process of the Bolthausen-Sznitman n-coalescent ever visits the state m is analyzed. It is shown that the asymptotic hitting probabilities h(m) = lim…

### Genealogies of regular exchangeable coalescents with applications to sampling

- Mathematics
- 2010

This article considers a model of genealogy corresponding to a regular exchangeable coalescent started from a large finite configuration, and undergoing neutral mutations, and derived analogous results for the number of active mutation-free lineages and the combined lineage lengths.

### On Asymptotics of Exchangeable Coalescents with Multiple Collisions

- MathematicsJournal of Applied Probability
- 2008

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…

### 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…

## References

SHOWING 1-10 OF 29 REFERENCES

### 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…

### On the number of segregating sites for populations with large family sizes

- MathematicsAdvances in Applied Probability
- 2006

We present recursions for the total number, S n , of mutations in a sample of n individuals, when the underlying genealogical tree of the sample is modelled by a coalescent process with mutation rate…

### 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…

### Coalescents with multiple collisions

- Mathematics
- 1999

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…

### The general coalescent with asynchronous mergers of ancestral lines

- MathematicsJournal 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…

### On the contraction method with degenerate limit equation

- Mathematics
- 2004

A class of random recursive sequences (Yn) with slowly varying variances as arising for parameters of random trees or recursive algorithms leads after normalizations to degenerate limit equations of…

### Much Ado about Derrida's GREM

- Mathematics
- 2007

We provide a detailed analysis of Derrida’s generalised random energy model (GREM). In particular, we describe its limiting Gibbs measure in terms of Ruelle’s Poisson cascades. Next we introduce and…

### Two Coalescents Derived from the Ranges of Stable Subordinators

- Mathematics
- 2000

Let $M_\alpha$ be the closure of the range of a stable subordinator of index $\alpha\in ]0,1[$. There are two natural constructions of the $M_{\alpha}$'s simultaneously for all $\alpha\in ]0,1[$, so…

### The Bolthausen–Sznitman coalescent and the genealogy of continuous-state branching processes

- Mathematics
- 2000

Abstract. We use Bochner’s subordination to give a representation of the genealogical structure associated with general continuous-state branching processes. We then apply this representation to…