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

Simatos, Florian and Schertzer, Emmanuel Height and contour processes of Crump-Mode-Jagers forests (I): general distribution and scaling limits in the case of short edges. (2018) Electronic Journal of Probability, 23 (67). 1-43. ISSN 1083-6489

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Official URL: http://doi.org/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.

Item Type:Article
Audience (journal):International peer-reviewed journal
Uncontrolled Keywords:
Institution:French research institutions > Centre National de la Recherche Scientifique - CNRS (FRANCE)
Université de Toulouse > Institut Supérieur de l'Aéronautique et de l'Espace - ISAE-SUPAERO (FRANCE)
Other partners > Université de Paris Diderot - Paris 7 (FRANCE)
Other partners > Université Pierre et Marie Curie, Paris 6 - UPMC (FRANCE)
Laboratory name:
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Deposited By: Florian Simatos
Deposited On:27 Nov 2017 14:58

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