Annales de l'Institut Henri Poincaré, Probabilités et Statistiques

Decomposition of Lévy trees along their diameter

Thomas Duquesne and Minmin Wang

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We study the diameter of Lévy trees that are random compact metric spaces obtained as the scaling limits of Galton–Watson trees. Lévy trees have been introduced by Le Gall & Le Jan (Ann. Probab. 26 (1998) 213–252) and they generalise Aldous’ Continuum Random Tree (1991) that corresponds to the Brownian case. We first characterize the law of the diameter of Lévy trees and we prove that it is realized by a unique pair of points. We prove that the law of Lévy trees conditioned to have a fixed diameter $r\in (0,\infty)$ is obtained by glueing at their respective roots two independent size-biased Lévy trees conditioned to have height $r/2$ and then by uniformly re-rooting the resulting tree; we also describe by a Poisson point measure the law of the subtrees that are grafted on the diameter. As an application of this decomposition of Lévy trees according to their diameter, we characterize the joint law of the height and the diameter of stable Lévy trees conditioned by their total mass; we also provide asymptotic expansions of the law of the height and of the diameter of such normalised stable trees, which generalises the identity due to Szekeres (In Combinatorial Mathematics, X (Adelaide, 1982) (1983) 392–397 Springer) in the Brownian case.


Nous étudions le diamètre des arbres de Lévy qui sont des espaces métriques compacts obtenus comme limites d’échelle des arbres de Galton–Watson. Les arbres de Lévy ont été introduits par Le Gall & Le Jan (Ann. Probab. 26 (1998) 213–252) et ils généralisent le Continuum Random Tree (1991) d’Aldous qui correspond au cas brownien. Nous caractérisons d’abord la loi du diamètre des arbres de Lévy et nous prouvons qu’une unique paire de points le réalise. Nous prouvons ensuite que la loi des arbres de Lévy conditionnés à avoir leur diamètre égal à $r\in{} ]0,\infty[$ est obtenu en collant à leurs racines respectives deux arbres de Lévy indépendants conditionnés chacuns à avoir une hauteur égale à $r/2$, et à réenraciner uniformément au hasard l’arbre obtenu par ce collage ; nous décrivons également en termes d’une mesure ponctuelle de Poisson, la loi des sous-arbres qui sont attachés le long du diamètre. En application de cette décomposition des arbres de Lévy le long de leur diamètre, nous caractérisons la loi jointe de la hauteur et du diamètre des arbres de Lévy stables conditionnés à avoir une masse totale unité. Nous donnons aussi des développements asymptotiques des lois de la hauteur et du diamètre de ces arbres stables normalisés, ce qui généralise une identité due à Szekeres (In Combinatorial Mathematics, X (Adelaide, 1982) (1983) 392–397 Springer) dans le cas brownien.

Article information

Ann. Inst. H. Poincaré Probab. Statist., Volume 53, Number 2 (2017), 539-593.

Received: 10 April 2015
Revised: 20 October 2015
Accepted: 21 October 2015
First available in Project Euclid: 11 April 2017

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60J80: Branching processes (Galton-Watson, birth-and-death, etc.) 60E07: Infinitely divisible distributions; stable distributions
Secondary: 60E10: Characteristic functions; other transforms 60G52: Stable processes 60G55: Point processes

Lévy trees Height process Diameter Decomposition Asymptotic expansion Stable law


Duquesne, Thomas; Wang, Minmin. Decomposition of Lévy trees along their diameter. Ann. Inst. H. Poincaré Probab. Statist. 53 (2017), no. 2, 539--593. doi:10.1214/15-AIHP725.

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  • [1] R. Abraham and J.-F. Delmas. Asymptotics for the small fragments of the fragmentation at nodes. Bernoulli 13 (1) (2007) 211–228.
  • [2] R. Abraham and J.-F. Delmas. Fragmentation associated with Lévy processes using snake. Probab. Theory Related Fields 141 (1–2) (2008) 113–154.
  • [3] R. Abraham and J.-F. Delmas. Williams’ decomposition of the Lévy continuum random tree and simultaneous extinction probability for populations with neutral mutations. Stochastic Process. Appl. 119 (4) (2009) 1124–1143.
  • [4] R. Abraham, J.-F. Delmas and G. Voisin. Pruning a Lévy continuum random tree. Electron. J. Probab. 15 (46) (2010) 1429–1473.
  • [5] D. Aldous. The continuum random tree. I. Ann. Probab. 19 (1) (1991) 1–28.
  • [6] D. Aldous. The continuum random tree. II. An overview. In Stochastic Analysis (Durham, 1990) 23–70. London Math. Soc. Lecture Note Ser. 167. Cambridge University Press, Cambridge, 1991.
  • [7] D. Aldous. The continuum random tree. III. Ann. Probab. 21 (1) (1993) 248–289.
  • [8] J. Bertoin. Lévy Processes. Cambridge Tracts in Mathematics 121. Cambridge University Press, Cambridge, 1996.
  • [9] J. M. Chambers, C. L. Mallows and B. W. Stuck. A method for simulating stable random variables. J. Amer. Statist. Assoc. 71 (354) (1976) 340–344.
  • [10] J. Dieudonné. Infinitesimal Calculus. Hermann, Paris; Houghton Mifflin, Boston, 1971. Translated from the French.
  • [11] A. Dress, V. Moulton and W. Terhalle. $T$-theory: An overview. European J. Combin. 17 (2–3) (1996) 161–175. In Discrete Metric Spaces (Bielefeld, 1994).
  • [12] T. Duquesne. A limit theorem for the contour process of conditioned Galton–Watson trees. Ann. Probab. 31 (2) (2003) 996–1027.
  • [13] T. Duquesne. The coding of compact real trees by real valued functions, 2006. Available at arXiv:math/0604106 [math.PR].
  • [14] T. Duquesne and J.-F. Le Gall. Random trees, Lévy processes and spatial branching processes. Astérisque 281 (2002) vi$+$147 pp.
  • [15] T. Duquesne and J.-F. Le Gall. Probabilistic and fractal aspects of Lévy trees. Probab. Theory Related Fields 131 (4) (2005) 553–603.
  • [16] T. Duquesne and J.-F. Le Gall. On the re-rooting invariance property of Lévy trees. Electron. Commun. Probab. 14 (2009) 317–326.
  • [17] T. Duquesne and M. Winkel. Growth of Lévy trees. Probab. Theory Related Fields 139 (3–4) (2007) 313–371.
  • [18] S. N. Evans. Probability and Real Trees. Lectures from the 35th Summer School on Probability Theory Held in Saint-Flour, July 6–23, 2005. Lecture Notes in Mathematics 1920. Springer, Berlin, 2008.
  • [19] S. N. Evans, J. Pitman and A. Winter. Rayleigh processes, real trees, and root growth with re-grafting. Probab. Theory Related Fields 134 (1) (2006) 81–126.
  • [20] C. Goldschmidt and B. Haas. Behavior near the extinction time in self-similar fragmentations. I. The stable case. Ann. Inst. Henri Poincaré Probab. Stat. 46 (2) (2010) 338–368.
  • [21] I. S. Gradshteyn and I. M. Ryzhik. Table of Integrals, Series, and Products, 7th edition. Elsevier/Academic Press, Amsterdam, 2007. Translated from the Russian, Translation edited and with a preface by Alan Jeffrey and Daniel Zwillinger, With one CD-ROM (Windows, Macintosh and UNIX).
  • [22] B. Haas and G. Miermont. The genealogy of self-similar fragmentations with negative index as a continuum random tree. Electron. J. Probab. 9 (4) (2004) 57–97 (electronic).
  • [23] E. Hille. Ordinary Differential Equations in the Complex Domain. Pure and Applied Mathematics. Wiley-Interscience [Wiley], New York–London–Sydney, 1976.
  • [24] Y. Hu and Z. Shi. Extreme lengths in Brownian and Bessel excursions. Bernoulli 3 (4) (1997) 387–402.
  • [25] I. A. Ibragimov and K. E. Černin. On the unimodality of stable laws. Teor. Veroyatn. Primen. 4 (1959) 453–456.
  • [26] I. Kortchemski. Invariance principles for Galton–Watson trees conditioned on the number of leaves. Stochastic Process. Appl. 122 (9) (2012) 3126–3172.
  • [27] J.-F. Le Gall. Spatial Branching Processes, Random Snakes and Partial Differential Equations. Lectures in Mathematics ETH Zürich. Birkhäuser, Basel, 1999.
  • [28] J.-F. Le Gall and Y. Le Jan. Branching processes in Lévy processes: The exploration process. Ann. Probab. 26 (1) (1998) 213–252.
  • [29] P. Marchal. A note on the fragmentation of a stable tree. In Fifth Colloquium on Mathematics and Computer Science 489–499. Discrete Math. Theor. Comput. Sci. Proc., AI, Assoc. Discrete Math. Theor. Comput. Sci., Nancy, 2008.
  • [30] G. Miermont. Self-similar fragmentations derived from the stable tree. I. Splitting at heights. Probab. Theory Related Fields 127 (3) (2003) 423–454.
  • [31] G. Miermont. Self-similar fragmentations derived from the stable tree. II. Splitting at nodes. Probab. Theory Related Fields 131 (3) (2005) 341–375.
  • [32] J. Pitman. Combinatorial stochastic processes. In Lectures from the 32nd Summer School on Probability Theory Held in Saint-Flour, July 7–24, 2002 7–24. Lecture Notes in Mathematics 1875. Springer, Berlin, 2006. With a foreword by Jean Picard.
  • [33] D. Revuz and M. Yor. Continuous Martingales and Brownian Motion, 3rd edition. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] 293. Springer, Berlin, 1999.
  • [34] G. Szekeres. Distribution of labelled trees by diameter. In Combinatorial Mathematics, X (Adelaide, 1982) 392–397. Lecture Notes in Mathematics 1036. Springer, Berlin, 1983.
  • [35] G. Tenenbaum. Introduction to Analytic and Probabilistic Number Theory. Cambridge Studies in Advanced Mathematics 46. Cambridge University Press, Cambridge, 1995. Translated from the second French edition (1995) by C. B. Thomas.
  • [36] M. Wang. Height and diameter of Brownian tree. Electron. Commun. Probab. 20 (88) (2015) 1–15.
  • [37] D. V. Widder. The Laplace Transform. Princeton Mathematical Series 6. Princeton University Press, Princeton, 1941.
  • [38] V. M. Zolotarev. One-Dimensional Stable Distributions. B. Silver (Ed.). Translations of Mathematical Monographs 65. Am. Math. Soc., Providence, 1986. Translated from the Russian by H. H. McFaden.