Electronic Journal of Probability

Height and contour processes of Crump-Mode-Jagers forests (I): general distribution and scaling limits in the case of short edges

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 are interested in the height and contour processes encoding a general CMJ tree.

We show that the one-dimensional distribution of the height process can be expressed in terms of a random transformation of the ladder height process associated with the underlying Lukasiewicz path. As an application of this result, when edges of the tree are “short” we show that, asymptotically, (1) the height process is obtained by stretching by a constant factor the height process of the associated genealogical Galton–Watson tree, (2) the contour process is obtained from the height process by a constant time change and (3) the CMJ trees converge in the sense of finite-dimensional distributions.

Article information

Source
Electron. J. Probab., Volume 23 (2018), paper no. 67, 43 pp.

Dates
Received: 22 July 2016
Accepted: 14 February 2018
First available in Project Euclid: 26 July 2018

Permanent link to this document
https://projecteuclid.org/euclid.ejp/1532570595

Digital Object Identifier
doi:10.1214/18-EJP151

Mathematical Reviews number (MathSciNet)
MR3835473

Zentralblatt MATH identifier
06924679

Subjects
Primary: 60J80: Branching processes (Galton-Watson, birth-and-death, etc.) 60J85: Applications of branching processes [See also 92Dxx] 60J70: Applications of Brownian motions and diffusion theory (population genetics, absorption problems, etc.) [See also 92Dxx] 60J75: Jump processes

Keywords
Crump-Mode-Jagers branching processes Contour processes Snakes Bellman-Harris processes scaling limits

Rights
Creative Commons Attribution 4.0 International License.

Citation

Schertzer, Emmanuel; Simatos, Florian. Height and contour processes of Crump-Mode-Jagers forests (I): general distribution and scaling limits in the case of short edges. Electron. J. Probab. 23 (2018), paper no. 67, 43 pp. doi:10.1214/18-EJP151. https://projecteuclid.org/euclid.ejp/1532570595


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