The Annals of Probability

The contour of splitting trees is a Lévy process

Amaury Lambert

Full-text: Open access


Splitting trees are those random trees where individuals give birth at a constant rate during a lifetime with general distribution, to i.i.d. copies of themselves. The width process of a splitting tree is then a binary, homogeneous Crump–Mode–Jagers (CMJ) process, and is not Markovian unless the lifetime distribution is exponential (or a Dirac mass at {∞}). Here, we allow the birth rate to be infinite, that is, pairs of birth times and life spans of newborns form a Poisson point process along the lifetime of their mother, with possibly infinite intensity measure.

A splitting tree is a random (so-called) chronological tree. Each element of a chronological tree is a (so-called) existence point (v, τ) of some individual v (vertex) in a discrete tree where τ is a nonnegative real number called chronological level (time). We introduce a total order on existence points, called linear order, and a mapping φ from the tree into the real line which preserves this order. The inverse of φ is called the exploration process, and the projection of this inverse on chronological levels the contour process.

For splitting trees truncated up to level τ, we prove that a thus defined contour process is a Lévy process reflected below τ and killed upon hitting 0. This allows one to derive properties of (i) splitting trees: conceptual proof of Le Gall–Le Jan’s theorem in the finite variation case, exceptional points, coalescent point process and age distribution; (ii) CMJ processes: one-dimensional marginals, conditionings, limit theorems and asymptotic numbers of individuals with infinite versus finite descendances.

Article information

Ann. Probab., Volume 38, Number 1 (2010), 348-395.

First available in Project Euclid: 25 January 2010

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60J80: Branching processes (Galton-Watson, birth-and-death, etc.)
Secondary: 37E25: Maps of trees and graphs 60G51: Processes with independent increments; Lévy processes 60G55: Point processes 60G70: Extreme value theory; extremal processes 60J55: Local time and additive functionals 60J75: Jump processes 60J85: Applications of branching processes [See also 92Dxx] 92D25: Population dynamics (general)

Real trees population dynamics contour process exploration process Poisson point process Crump–Mode–Jagers branching process Malthusian parameter Lévy process scale function composition of subordinators Jirina process coalescent point process limit theorems Yaglom distribution modified geometric distribution


Lambert, Amaury. The contour of splitting trees is a Lévy process. Ann. Probab. 38 (2010), no. 1, 348--395. doi:10.1214/09-AOP485.

Export citation


  • [1] Aldous, D. (1991). The continuum random tree. I. Ann. Probab. 19 1–28.
  • [2] Aldous, D. (1993). The continuum random tree. III. Ann. Probab. 21 248–289.
  • [3] Bertoin, J. (1996). Lévy Processes. Cambridge Tracts in Mathematics 121. Cambridge Univ. Press, Cambridge.
  • [4] Bertoin, J., Fontbona, J. and Martínez, S. (2008). On prolific individuals in a supercritical continuous-state branching process. J. Appl. Probab. 45 714–726.
  • [5] Bertoin, J. and Le Gall, J.-F. (2000). The Bolthausen–Sznitman coalescent and the genealogy of continuous-state branching processes. Probab. Theory Related Fields 117 249–266.
  • [6] Dress, A. W. M. and Terhalle, W. F. (1996). The real tree. Adv. Math. 120 283–301.
  • [7] Duquesne, T. (2007). The coding of compact real trees by real valued functions. Preprint. Available at arXiv PR/0604106.
  • [8] Duquesne, T. and Le Gall, J. F. (2002). Random trees, Lévy processes and spatial branching processes. Astérisque 281 1–147.
  • [9] Evans, S. N. (2009). Probability and Real Trees. Lecture Notes in Math. 1920. Springer, Berlin.
  • [10] Geiger, J. and Kersting, G. (1997). Depth-first search of random trees, and Poisson point processes. In Classical and Modern Branching Processes (Minneapolis, MN, 1994). IMA Math. Appl. 84 111–126. Springer, New York.
  • [11] Jaffard, S. (1999). The multifractal nature of Lévy processes. Probab. Theory Related Fields 114 207–227.
  • [12] Jiřina, M. (1958). Stochastic branching processes with continuous state space. Czechoslovak Math. J. 8 292–313.
  • [13] Lambert, A. (2002). The genealogy of continuous-state branching processes with immigration. Probab. Theory Related Fields 122 42–70.
  • [14] Lambert, A. (2003). Coalescence times for the branching process. Adv. in Appl. Probab. 35 1071–1089.
  • [15] Lambert, A. (2008). Population dynamics and random genealogies. Stoch. Models 24 45–163.
  • [16] Lambert, A. (2009). The allelic partition for coalescent point processes. Markov Processes Relat. Fields. Preprint. To appear. Available at arXiv:0804.2572v2.
  • [17] Lambert, A. (2010). Spine decompositions of Lévy trees. To appear. In preparation.
  • [18] Le Gall, J.-F. (1993). The uniform random tree in a Brownian excursion. Probab. Theory Related Fields 96 369–383.
  • [19] Le Gall, J.-F. (2005). Random trees and applications. Probab. Surv. 2 245–311 (electronic).
  • [20] Le Gall, J.-F. and Le Jan, Y. (1998). Branching processes in Lévy processes: The exploration process. Ann. Probab. 26 213–252.
  • [21] Neveu, J. (1986). Arbres et processus de Galton–Watson. Ann. Inst. H. Poincaré Probab. Statist. 22 199–207.
  • [22] Nerman, O. (1981). On the convergence of supercritical general (C–M–J) branching processes. Z. Wahrsch. Verw. Gebiete 57 365–395.
  • [23] O’Connell, N. (1995). The genealogy of branching processes and the age of our most recent common ancestor. Adv. in Appl. Probab. 27 418–442.
  • [24] Popovic, L. (2004). Asymptotic genealogy of a critical branching process. Ann. Appl. Probab. 14 2120–2148.
  • [25] Revuz, D. and Yor, M. (1999). Continuous Martingales and Brownian Motion, 3rd ed. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] 293. Springer, Berlin.
  • [26] Taïb, Z. (1992). Branching Processes and Neutral Evolution. Lecture Notes in Biomathematics 93. Springer, Berlin.