Sub-exponential tail bounds for conditioned stable Bienaymé–Galton–Watson trees

  • Igor KortchemskiEmail author


We establish uniform sub-exponential tail bounds for the width, height and maximal outdegree of critical Bienaymé–Galton–Watson trees conditioned on having a large fixed size, whose offspring distribution belongs to the domain of attraction of a stable law. This extends results obtained for the height and width by Addario-Berry, Devroye and Janson in the finite variance case.


Random trees Bienaymé–Galton–Watson trees Spectrally positive stable Lévy processes Non-crossing trees 

Mathematics Subject Classification

Primary 60J80 05C05 05C07 Secondary 60F05 60G52 



The author is grateful to Louigi Addario-Berry and to Svante Janson for stimulating discussions, as well as to an anonymous referee for her or his extremely careful reading and many comments that greatly improved the paper, and would like to thank the Newton Institute, where this work was finalized, for hospitality.


  1. 1.
    Addario-Berry, L.: Tail bounds for the height and width of a random tree with a given degree sequence. Random Struct. Algorithms 41, 253–261 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Addario-Berry, L., Devroye, L., Janson, S.: Sub-Gaussian tail bounds for the width and height of conditioned Galton-Watson trees. Ann. Probab. 41, 1072–1087 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Bertoin, J.: Lévy Processes, vol. 121 of Cambridge Tracts in Mathematics. Cambridge University Press, Cambridge (1996)Google Scholar
  4. 4.
    Bertoin, J.: On the maximal offspring in a critical branching process with infinite variance. J. Appl. Probab. 48, 576–582 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Bertoin, J.: On largest offspring in a critical branching process with finite variance. J. Appl. Probab. 50, 791–800 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Bhamidi, S., van der Hofstad, R., Sen, S.: The multiplicative coalescent, inhomogeneous continuum random trees, and new universality classes for critical random graphs. Preprint available on arxiv, arXiv:1508.04645
  7. 7.
    Bingham, N.H., Goldie, C.M., Teugels, J.L.: Regular Variation, vol. 27 of Encyclopedia of Mathematics and its Applications. Cambridge University Press, Cambridge (1987)zbMATHGoogle Scholar
  8. 8.
    Björnberg, J.E., Stefánsson, S.Ö.: Random walk on random infinite looptrees. J. Stat. Phys. 158, 1234–1261 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Broutin, N., Marckert, J.-F.: Asymptotics of trees with a prescribed degree sequence and applications. Random Struct. Algorithms 44, 290–316 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Chaumont, L.: Excursion normalisée, méandre et pont pour les processus de Lévy stables. Bull. Sci. Math. 121, 377–403 (1997)MathSciNetzbMATHGoogle Scholar
  11. 11.
    Croydon, D., Kumagai, T.: Random walks on Galton-Watson trees with infinite variance offspring distribution conditioned to survive. Electron. J. Probab. 13(51), 1419–1441 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Curien, N., Haas, B., Kortchemski, I.: The CRT is the scaling limit of random dissections. Random Struct. Algorithms 47, 304–327 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Curien, N., Kortchemski, I.: Random non-crossing plane configurations: a conditioned Galton-Watson tree approach. Random Struct. Algorithms 45, 236–260 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Curien, N., Kortchemski, I.: Random stable looptrees. Electron. J. Probab. 19(108), 35 (2014)MathSciNetzbMATHGoogle Scholar
  15. 15.
    Curien, N., Kortchemski, I.: Percolation on random triangulations and stable looptrees. Probab. Theory Relat. Fields 163, 303–337 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Deutsch, E., Noy, M.: Statistics on non-crossing trees. Discrete Math. 254, 75–87 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Duplantier, B., Miller, J., Sheffield, S.: Liouville quantum gravity as a mating of trees. Preprint available on arxiv, arXiv:1409.7055
  18. 18.
    Duquesne, T.: A limit theorem for the contour process of conditioned Galton-Watson trees. Ann. Probab. 31, 996–1027 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Duquesne, T., Wang, M.: Decomposition of Lévy trees along their diameter. To appear in Ann. Inst. H. Poincaré Probab. StatistGoogle Scholar
  20. 20.
    Feller, W.: An Introduction to Probability Theory and Its Applications, vol. II, 2nd edn. John Wiley & Sons Inc., New York (1971)zbMATHGoogle Scholar
  21. 21.
    Haas, B., Miermont, G.: Scaling limits of Markov branching trees, with applications to Galton-Watson and random unordered trees. Ann. Probab. 40, 2589–2666 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Ibragimov, I.A., Linnik, Y.V.: Independent and stationary sequences of random variables. Wolters-Noordhoff Publishing, Groningen (1971). With a supplementary chapter by I. A. Ibragimov and V. V. Petrov, Translation from the Russian edited by J. F. C. KingmanGoogle Scholar
  23. 23.
    Jacod, J., Shiryaev, A.N.: Limit Theorems for Stochastic Processes, vol. 288 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 2nd edn. Springer-Verlag, Berlin (2003)Google Scholar
  24. 24.
    Janson, S.: Random cutting and records in deterministic and random trees. Random Struct. Algorithms 29, 139–179 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Janson, S.: Simply generated trees, conditioned Galton-Watson trees, random allocations and condensation. Probab. Surv. 9, 103–252 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Kersting, G.: On the Height Profile of a Conditioned Galton-Watson Tree. Unpublished manuscriptGoogle Scholar
  27. 27.
    Kesten, H.: Subdiffusive behavior of random walk on a random cluster. Ann. Inst. H. Poincaré Probab. Stat. 22, 425–487 (1986)MathSciNetzbMATHGoogle Scholar
  28. 28.
    Kingman, J.F.C.: Uses of exchangeability. Ann. Probab. 6, 183–197 (1978)MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Kortchemski, I.: Invariance principles for Galton-Watson trees conditioned on the number of leaves. Stoch. Process. Appl. 122, 3126–3172 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Kortchemski, I.: A simple proof of Duquesne’s theorem on contour processes of conditioned Galton-Watson trees. In: Séminaire de Probabilités XLV, vol. 2078 of Lecture Notes in Math., pp. 537–558. Springer, Cham (2013)Google Scholar
  31. 31.
    Kortchemski, I., Marzouk, C.: Triangulating stable laminations. Electron. J. Probab. 21(11), 1–31 (2016)MathSciNetzbMATHGoogle Scholar
  32. 32.
    Le Gall, J.-F.: Random trees and applications. Probability Surveys (2005)Google Scholar
  33. 33.
    Le Gall, J.-F., Le Jan, Y.: Branching processes in Lévy processes: the exploration process. Ann. Probab. 26, 213–252 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  34. 34.
    Le Gall, J.-F., Miermont, G.: Scaling limits of random planar maps with large faces. Ann. Probab. 39, 1–69 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  35. 35.
    Lyons, R., Pemantle, R., Peres, Y.: Conceptual proofs of \({L}\) log \({L}\) criteria for mean behavior of branching processes. Ann. Probab. 23, 1125–1138 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  36. 36.
    Marckert, J.-F., Panholzer, A.: Noncrossing trees are almost conditioned Galton-Watson trees. Random Struct. Algorithms 20, 115–125 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  37. 37.
    Pakes, A.G.: Some new limit theorems for the critical branching process allowing immigration. Stoch. Process. Appl. 4, 175–185 (1976)MathSciNetCrossRefzbMATHGoogle Scholar
  38. 38.
    Pakes, A.G.: Extreme order statistics on Galton-Watson trees. Metrika 47, 95–117 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  39. 39.
    Panagiotou, K., Stufler, B., Weller, K.: Scaling Limits of Random Graphs from Subcritical Classes. To appear in Ann. ProbabGoogle Scholar
  40. 40.
    Pitman, J.: Combinatorial Stochastic Processes, vol. 1875 of Lecture Notes in Mathematics. Springer-Verlag, Berlin (2006). Lectures from the 32nd Summer School on Probability Theory held in Saint-Flour, July 7-24, 2002, With a foreword by Jean PicardGoogle Scholar
  41. 41.
    Rahimov, I., Yanev, G.P.: On maximum family size in branching processes. J. Appl. Probab. 36, 632–643 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  42. 42.
    Ray, G.: Large unicellular maps in high genus. Ann. Inst. Henri Poincaré Probab. Stat. 51, 1432–1456 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  43. 43.
    Seneta, E.: Regularly Varying Functions. Springer-Verlag, Berlin (1976)CrossRefzbMATHGoogle Scholar
  44. 44.
    Slack, R.S.: A branching process with mean one and possibly infinite variance. Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 9, 139–145 (1968)MathSciNetCrossRefzbMATHGoogle Scholar
  45. 45.
    Stufler, B.: Random enriched trees with applications to random graphs. Preprint available on arxiv, arXiv:1504.02006
  46. 46.
    Vatutin, V.A., Wachtel, V., Vitalihtel, Fleishcmann, K.: Critical Galton-Watson branching processes: the maximum of the total number of particles within a large window. Teor. Veroyatn. Primen. 52, 419–445 (2007)MathSciNetCrossRefGoogle Scholar
  47. 47.
    Whitt, W.: Stochastic-process limits, Springer Series in Operations Research. Springer-Verlag, New York (2002). An introduction to stochastic-process limits and their application to queuesGoogle Scholar
  48. 48.
    Zolotarev, V.M.: One-dimensional stable distributions, vol. 65 of Translations of Mathematical Monographs, vol. 65. American Mathematical Society, Providence, RI (1986). Translated from the Russian by H. H. McFaden, Translation edited by Ben SilverGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2016

Authors and Affiliations

  1. 1.CNRS and CMAPÉcole polytechniquePalaiseauFrance

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