The Annals of Probability

A limit theorem for the contour process of condidtioned Galton--Watson trees

Thomas Duquesne

Full-text: Open access


In this work, we study asymptotics of the genealogy of Galton--Watson processes conditioned on the total progeny. We consider a fixed, aperiodic and critical offspring distribution such that the rescaled Galton--Watson processes converges to a continuous-state branching process (CSBP) with a stable branching mechanism of index $\alpha \in (1, 2]$. We code the genealogy by two different processes: the contour process and the height process that Le Gall and Le Jan recently introduced. We show that the rescaled height process of the corresponding Galton--Watson family tree, with one ancestor and conditioned on the total progeny, converges in a functional sense, to a new process: the normalized excursion of the continuous height process associated with the $\alpha $-stable CSBP. We deduce from this convergence an analogous limit theorem for the contour process. In the Brownian case $\alpha =2$, the limiting process is the normalized Brownian excursion that codes the continuum random tree: the result is due to Aldous who used a different method.

Article information

Ann. Probab. Volume 31, Number 2 (2003), 996-1027.

First available in Project Euclid: 24 March 2003

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60F17: Functional limit theorems; invariance principles 05G05 60G52: Stable processes 60G17: Sample path properties

Stable continuous random tree limit theorem conditioned Galton--Watson tree


Duquesne, Thomas. A limit theorem for the contour process of condidtioned Galton--Watson trees. Ann. Probab. 31 (2003), no. 2, 996--1027. doi:10.1214/aop/1048516543.

Export citation


  • [1] ALDOUS, D. J. (1991). The continuum random tree I. Ann. Probab. 19 1-28.
  • [2] ALDOUS, D. J. (1993). The continuum random tree III. Ann. Probab. 21 248-289.
  • [3] BENNIES, J. and KERSTING, G. (2000). A random walk approach to Galton-Watson trees. J. Theoret. Probab. 13 777-803.
  • [4] BERTOIN, J. (1996). Lévy Processes. Cambridge Univ. Press.
  • [5] BINGHAM, N. H. (1976). Continuous branching processes and spectral positivity. Stochastic Process. Appl. 4 217-242.
  • [6] BINGHAM, N. H., GOLDIES, C. M. and TEUGELS, J. L. (1987). Regular Variation. Cambridge Univ. Press.
  • [7] CHAUMONT, L. (1994). Processus de Lévy et conditionnement. Thèse de doctorat, Laboratoire de Probabilités de Paris 6.
  • [8] CHAUMONT, L. (1997). Excursion normalisée, méandre et pont pour les processus de Lévy stables. Bull. Sci. Math. 121 377-403.
  • [9] DRESS, A. and TERHALLE, W. (1996). The real tree. Adv. in Math. 120 283-301.
  • [10] DRESS, A., MOULTON, V. and TERHALLE, W. (1996). T -theory: An overview. European Journal of Combinatorics 17 161-175.
  • [11] DUQUESNE, T. and LE GALL, J.-F. (2002). Random Trees, Lévy Processes and Spatial Branching Processes. Astérisque 281.
  • [12] GREY, D. R. (1974). Asy mptotic behaviour of continuous-time continuous state-space branching processes. J. Appl. Probab. 11 669-677.
  • [13] GRIMVALL, A. (1974). On the convergence of a sequence of branching processes. Ann. Probab. 2 1027-1045.
  • [14] JACOD, J. and SHIRy AEV, A. N. (1987). Limit Theorems for Stochastic Processes. Springer, Berlin.
  • [15] JIRINA, M. (1958). Stochastic branching processes with continous state-space. Czech. Math. J. 8 292-313.
  • [16] KERSTING, G. (1998). On the height profile of a conditioned Galton-Watson tree. Preprint.
  • [17] LAMPERTI, J. (1967). Continuous-state branching processes. Bull. Amer. Math. Soc. 73 382- 386.
  • [18] LAMPERTI, J. (1967). The limit of a sequence of branching processes. Z. Wahrsch. Verw. Gebiete 7 271-288.
  • [19] LAMPERTI, J. (1967). Limiting distributions of branching processes. Proc. Fifth Berkeley Sy mp. Math. Statist. Probab. 2 225-241. Univ. California Press, Berkeley.
  • [20] LE GALL, J.-F. (1993). The uniform random tree in a Brownian excursion. Probab. Theory Related Fields 96 369-383.
  • [21] LE GALL, J.-F. and LE JAN, Y. (1998). Branching processes in Lévy processes: The exploration process. Ann. Probab. 26 213-252.
  • [22] LE GALL, J.-F. and LE JAN, Y. (1998). Branching processes in Lévy processes: Laplace functionals of snakes and superprocesses. Ann. Probab. 26 1407-1432.
  • [23] LIMIC, V. (1999). A LIFO queue in heavy traffic. Preprint.
  • [24] NEVEU, J. (1986). Arbres et processus de Galton-Watson. Ann. Inst. H. Poincaré 26 199-207.
  • [25] OTTER, R. (1949). The multiplicative process. Ann. Math. Statist. 20 206-224.
  • [26] ROGERS, L. C. G. (1984). A new identity for real Lévy processes. Ann. Inst. H. Poincaré 20 21-34.
  • [27] VERVAAT, W. (1979). A relation between Brownian bridge and Brownian excursion. Ann. Probab. 7 141-149.