Electronic Journal of Probability

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

Emmanuel Schertzer and Florian Simatos

Full-text: Open access


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

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

Received: 3 September 2018
Accepted: 10 April 2019
First available in Project Euclid: 18 May 2019

Permanent link to this document

Digital Object Identifier

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

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

Creative Commons Attribution 4.0 International License.


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

Export citation


  • [1] Romain Abraham and Laurent Serlet. Poisson snake and fragmentation. Electron. J. Probab., 7(17): 15 pp. (electronic), 2002.
  • [2] David Aldous. The continuum random tree. III. Ann. Probab., 21(1):248–289, 1993.
  • [3] Jean Bertoin. Lévy processes, volume 121 of Cambridge Tracts in Mathematics. Cambridge University Press, Cambridge, 1996.
  • [4] N. H. Bingham, C. M. Goldie, and J. L. Teugels. Regular variation, volume 27 of Encyclopedia of Mathematics and its Applications. Cambridge University Press, Cambridge, 1989.
  • [5] Bertrand Cloez and Benoît Henry. Markovian tricks for non-Markovian trees: contour process, extinction and scaling limits. arXiv:1801.03284.
  • [6] Miraine Dávila Felipe and Amaury Lambert. Time reversal dualities for some random forests. ALEA Lat. Am. J. Probab. Math. Stat., 12(1):399–426, 2015.
  • [7] Cécile Delaporte. Lévy processes with marked jumps I: Limit theorems. J. Theoret. Probab., 28(4):1468–1499, 2015.
  • [8] Thomas Duquesne. The coding of compact real trees by real valued functions. arXiv:math/0604106.
  • [9] Thomas Duquesne and Jean-François Le Gall. Random trees, Lévy processes and spatial branching processes. Astérisque, (281):vi+147, 2002.
  • [10] Steven N. Evans. Probability and real trees, volume 1920 of Lecture Notes in Mathematics. Springer, Berlin, 2008. Lectures from the 35th Summer School on Probability Theory held in Saint-Flour, July 6–23, 2005.
  • [11] Bernhard Gittenberger. Convergence of branching processes to the local time of a Bessel process. In Proceedings of the Eighth International Conference “Random Structures and Algorithms” (Poznan, 1997), volume 13, pages 423–438, 1998.
  • [12] Bernhard Gittenberger. A note on: “State spaces of the snake and its tour—convergence of the discrete snake” [J. Theoret. Probab. 16(4):1015–1046 (2003); mr2033196]. by J.-F. Marckert and A. Mokkadem. J. Theoret. Probab., 16(4):1063–1067 (2004), 2003.
  • [13] Jean Jacod and Albert N. Shiryaev. Limit theorems for stochastic processes, volume 288 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]. Springer-Verlag, Berlin, second edition, 2003.
  • [14] Svante Janson and Jean-François Marckert. Convergence of discrete snakes. J. Theoret. Probab., 18(3):615–647, 2005.
  • [15] Olav Kallenberg. Foundations of modern probability. Probability and its Applications (New York). Springer-Verlag, New York, second edition, 2002.
  • [16] Thomas G. Kurtz. Random time changes and convergence in distribution under the Meyer-Zheng conditions. Ann. Probab., 19(3):1010–1034, 1991.
  • [17] Amaury Lambert. The contour of splitting trees is a Lévy process. Ann. Probab., 38(1):348–395, 2010.
  • [18] Amaury Lambert and Florian Simatos. Asymptotic Behavior of Local Times of Compound Poisson Processes with Drift in the Infinite Variance Case. J. Theoret. Probab., 28(1):41–91, 2015.
  • [19] Amaury Lambert, Florian Simatos, and Bert Zwart. Scaling limits via excursion theory: Interplay between Crump-Mode-Jagers branching processes and Processor-Sharing queues. Ann. Appl. Probab., 23(6):2357–2381, 2013.
  • [20] Jean-François Le Gall and Yves Le Jan. Branching processes in Lévy processes: the exploration process. Ann. Probab., 26(1):213–252, 1998.
  • [21] Jean-François Marckert and Abdelkader Mokkadem. The depth first processes of Galton-Watson trees converge to the same Brownian excursion. Ann. Probab., 31(3):1655–1678, 2003.
  • [22] Jean-François Marckert and Abdelkader Mokkadem. States spaces of the snake and its tour—convergence of the discrete snake. J. Theoret. Probab., 16(4):1015–1046 (2004), 2003.
  • [23] Mathieu Richard. Lévy processes conditioned on having a large height process. Ann. Inst. Henri Poincaré Probab. Stat., 49(4):982–1013, 2013.
  • [24] Mathieu Richard. Splitting trees with neutral mutations at birth. Stochastic Process. Appl., 124(10):3206–3230, 2014.
  • [25] Emmanuel Schertzer and Florian Simatos. Height and contour processes of Crump-Mode-Jagers forests (I): general distribution and scaling limits in the case of short edges. arXiv:1506.03192.
  • [26] V. A. Vatutin. A new limit theorem for a critical Bellman-Harris branching process. Mat. Sb. (N.S.), 109(151)(3):440–452, 480, 1979.
  • [27] Ward Whitt. Some useful functions for functional limit theorems. Math. Oper. Res., 5(1):67–85, 1980.