Electronic Journal of Probability

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

Emmanuel Schertzer and Florian Simatos

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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


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