# The size of the last merger and time reversal in $\Lambda$-coalescents

@article{Kersting2018TheSO, title={The size of the last merger and time reversal in \$\Lambda\$-coalescents}, author={G{\"o}tz Kersting and Jason Schweinsberg and A. Wakolbinger}, journal={Annales de l'Institut Henri Poincar{\'e}, Probabilit{\'e}s et Statistiques}, year={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 conditions under which, as $n \to \infty$, the sequence of these random variables a) is tight, b) converges in distribution to a finite random variable or c) converges to infinity in probability. Our conditions are optimal for $\Lambda$-coalescents that have a dust component. For general $\Lambda…

## 8 Citations

The collision spectrum of $\Lambda$-coalescents

- Physics, MathematicsThe Annals of Applied Probability
- 2018

$\Lambda$-coalescents model the evolution of a coalescing system in which any number of blocks randomly sampled from the whole may merge into a larger block. For the coalescent restricted to…

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

- Physics
- 2020

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.…

On the size of the block of 1 for $\varXi$-coalescents with dust

- Mathematics
- 2017

We study the frequency process $f_1$ of the block of 1 for a $\varXi$-coalescent $\varPi$ with dust. If $\varPi$ stays infinite, $f_1$ is a jump-hold process which can be expressed as a sum of broken…

External branch lengths of $\Lambda $-coalescents without a dust component

- MathematicsElectronic Journal of Probability
- 2019

$\Lambda$-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…

THE COLLISION SPECTRUM OF -COALESCENTS1

- Physics, Mathematics
- 2018

-coalescents model the evolution of a coalescing system in which any number of blocks randomly sampled from the whole may merge into a larger block. For the coalescent restricted to initially n…

On the time to absorption in $\Lambda$-coalescents

- Mathematics
- 2017

We present a law of large numbers and a central limit theorem for the time to absorption of $\Lambda$-coalescents, started from $n$ blocks, as $n \to \infty$. The proofs rely on an approximation of…

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…

The time to absorption in Λ-coalescents

- MathematicsAdvances in Applied Probability
- 2018

Abstract We present a law of large numbers and a central limit theorem for the time to absorption of Λ-coalescents with dust started from n blocks, as n→∞. The proofs rely on an approximation of the…

## References

SHOWING 1-10 OF 17 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…

$\beta$-coalescents and stable Galton-Watson trees

- Mathematics
- 2013

Representation of coalescent process using pruning of trees has been used by Goldschmidt and Martin for the Bolthausen-Sznitman coalescent and by Abraham and Delmas for the…

-coalescents and stable Galton-Watson trees

- Mathematics
- 2013

. Representation of coalescent process using pruning of trees has been used by Goldschmidt and Martin for the Bolthausen-Sznitman coalescent and by Abraham and Delmas for the β (3 / 2 , 1 /…

On hitting probabilities of beta coalescents and absorption times of coalescents that come down from infinity

- Mathematics
- 2014

Let X = (Xk)k=0;1;::: denote the jump chain of the block counting process of the -coalescent with = (2 ; ) being the beta distribution with parameter 2 (0;2). A solution for the hitting probability…

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 fixation line in the Λ -coalescent.

- Mathematics
- 2015

We define a Markov process in a forward population model with backward genealogy given by the -coalescent. This Markov process, called the fixation line, is related to the block counting process…

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 Λ-Coalescents with Dust Component

- Mathematics, PhysicsJournal of Applied Probability
- 2011

We consider the Λ-coalescent processes with a positive frequency of singleton clusters. The class in focus covers, for instance, the beta(a, b)-coalescents with a > 1. We show that some large-sample…

A NECESSARY AND SUFFICIENT CONDITION FOR THE Λ-COALESCENT TO COME DOWN FROM INFINITY

- Mathematics

Let Λ be a finite measure on the Borel subsets of [0, 1]. Let Π∞ be the standard Λ-coalescent, which is defined in [4] and also studied in [5]. Then Π∞ is a Markov process whose state space is the…

A Construction of a β-Coalescent via the Pruning of Binary Trees

- Computer Science, MathematicsJournal of Applied Probability
- 2013

Considering a random binary tree with n labelled leaves, we use a pruning procedure on this tree in order to construct a β(3/2,1/2)-coalescent process. We also use the continuous analogue of this…