## Electronic Journal of Probability

- Electron. J. Probab.
- Volume 24 (2019), paper no. 47, 38 pp.

### Height and contour processes of Crump-Mode-Jagers forests (II): the Bellman–Harris universality class

Emmanuel Schertzer and Florian Simatos

#### Abstract

Crump–Mode–Jagers (CMJ) trees generalize Galton–Watson trees by allowing individuals to live for an arbitrary duration and give birth at arbitrary times during their life-time. In this paper, we exhibit a simple condition under which the height and contour processes of CMJ forests belong to the universality class of Bellman–Harris processes. This condition formalizes an asymptotic independence between the chronological and genealogical structures. We show that it is satisfied by a large class of CMJ processes and in particular, quite surprisingly, by CMJ processes with a finite variance offspring distribution. Along the way, we prove a general tightness result.

#### Article information

**Source**

Electron. J. Probab., Volume 24 (2019), paper no. 47, 38 pp.

**Dates**

Received: 3 September 2018

Accepted: 10 April 2019

First available in Project Euclid: 18 May 2019

**Permanent link to this document**

https://projecteuclid.org/euclid.ejp/1558145015

**Digital Object Identifier**

doi:10.1214/19-EJP307

**Subjects**

Primary: 60F17: Functional limit theorems; invariance principles 60J80: Branching processes (Galton-Watson, birth-and-death, etc.)

**Keywords**

Crump-Mode-Jagers branching processes chronologial trees scaling limits invariance principles

**Rights**

Creative Commons Attribution 4.0 International License.

#### Citation

Schertzer, Emmanuel; Simatos, Florian. Height and contour processes of Crump-Mode-Jagers forests (II): the Bellman–Harris universality class. Electron. J. Probab. 24 (2019), paper no. 47, 38 pp. doi:10.1214/19-EJP307. https://projecteuclid.org/euclid.ejp/1558145015