# On Λ-Coalescents with Dust Component

@article{Gnedin2011OnW, title={On $\Lambda$-Coalescents with Dust Component}, author={Alexander V. Gnedin and Alexander Iksanov and Alexander Marynych}, journal={Journal of Applied Probability}, year={2011}, volume={48}, pages={1133 - 1151} }

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 properties of these processes can be derived by coupling the coalescent with an increasing Lévy process (subordinator), and by exploiting parallels with the theory of regenerative composition structures. In particular, we discuss the limit distributions of the absorption time and the number of…

## 26 Citations

### 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…

### Λ-coalescents: a survey

- PhysicsJournal of Applied Probability
- 2014

Λ-coalescents model the evolution of a coalescing system in which any number of components randomly sampled from the whole may merge into larger blocks. This survey focuses on related combinatorial…

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

- Mathematics
- 2017

We consider the number of blocks involved in the last merger of a Λ-coalescent started with n blocks. We give conditions under which, as n → ∞ , the sequence of these random variables a) is tight, b)…

### On Asymptotics of the Beta Coalescents

- MathematicsAdvances in Applied Probability
- 2014

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…

### 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…

### The rate of convergence of the block counting process of exchangeable coalescents with dust

- MathematicsLatin American Journal of Probability and Mathematical Statistics
- 2021

For exchangeable coalescents with dust the rate of convergence as the sample size tends to infinity of the scaled block counting process to the frequency of singleton process is determined. This rate…

### 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…

### On the block counting process and the fixation line of exchangeable coalescents

- Mathematics
- 2016

We study the block counting process and the fixation line of exchangeable coalescents. Formulas for the infinitesimal rates of both processes are provided. It is shown that the block counting process…

### A survey of the -coalescents

- Physics
- 2014

coalescents model evolution of a coalescing system in which any number of com- ponents randomly sampled from the whole may merge in larger blocks. This survey focuses on related combinatorial…

### Scaling limits for the block counting process and the fixation line for a class of Λ-coalescents

- MathematicsLatin American Journal of Probability and Mathematical Statistics
- 2022

. We provide scaling limits for the block counting process and the ﬁxation line of Λ coalescents as the initial state n tends to inﬁnity under the assumption that the measure Λ on [0 , 1] satisﬁes…

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