## Electronic Communications in Probability

- Electron. Commun. Probab.
- Volume 22 (2017), paper no. 48, 12 pp.

### Asymptotic number of caterpillars of regularly varying $\Lambda $-coalescents that come down from infinity

#### Abstract

In this paper we look at the asymptotic number of $r$-caterpillars for $\Lambda $-coalescents which come down from infinity, under a regularly varying assumption. An $r$-caterpillar is a functional of the coalescent process started from $n$ individuals which, roughly speaking, is a block of the coalescent at some time, formed by one line of descend to which $r-1$ singletons have merged one by one. We show that the number of $r$-caterpillars, suitably scaled, converge to an explicit constant as the sample size $n$ goes to $\infty $.

#### Article information

**Source**

Electron. Commun. Probab., Volume 22 (2017), paper no. 48, 12 pp.

**Dates**

Received: 7 December 2016

Accepted: 28 August 2017

First available in Project Euclid: 2 October 2017

**Permanent link to this document**

https://projecteuclid.org/euclid.ecp/1506931448

**Digital Object Identifier**

doi:10.1214/17-ECP81

**Mathematical Reviews number (MathSciNet)**

MR3710804

**Zentralblatt MATH identifier**

06797801

**Subjects**

Primary: 60J80: Branching processes (Galton-Watson, birth-and-death, etc.) 60J99: None of the above, but in this section 60F99: None of the above, but in this section

**Keywords**

coalescent processes regularly varying coalescents cherries caterpillars scaling limits

**Rights**

Creative Commons Attribution 4.0 International License.

#### Citation

Şengül, Batı. Asymptotic number of caterpillars of regularly varying $\Lambda $-coalescents that come down from infinity. Electron. Commun. Probab. 22 (2017), paper no. 48, 12 pp. doi:10.1214/17-ECP81. https://projecteuclid.org/euclid.ecp/1506931448