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2018 Height and contour processes of Crump-Mode-Jagers forests (I): general distribution and scaling limits in the case of short edges
Emmanuel Schertzer, Florian Simatos
Electron. J. Probab. 23: 1-43 (2018). DOI: 10.1214/18-EJP151

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.

Citation

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Emmanuel Schertzer. Florian Simatos. "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 1 - 43, 2018. https://doi.org/10.1214/18-EJP151

Information

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

zbMATH: 06924679
MathSciNet: MR3835473
Digital Object Identifier: 10.1214/18-EJP151

Subjects:
Primary: 60J70, 60J75, 60J80, 60J85

JOURNAL ARTICLE
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