Electronic Communications in Probability

An elementary proof of Hawkes's conjecture on Galton-Watson trees.

Thomas Duquesne

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Abstract

In 1981, J. Hawkes conjectured the exact form of the Hausdorff gauge function for the boundary of supercritical Galton-Watson trees under a certain assumption on the tail at infinity of the total mass of the branching measure. Hawkes's conjecture has been proved by T. Watanabe in 2007 as well as other precise results on fractal properties of the boundary of Galton-Watson trees. The goal of this paper is to provide an elementary proof of Hawkes's conjecture under a less restrictive assumption than in T. Watanabe's paper, by use of size-biased Galton-Watson trees introduced by Lyons, Pemantle and Peres in 1995.

Article information

Source
Electron. Commun. Probab., Volume 14 (2009), paper no. 15, 151-164.

Dates
Accepted: 19 April 2009
First available in Project Euclid: 6 June 2016

Permanent link to this document
https://projecteuclid.org/euclid.ecp/1465234724

Digital Object Identifier
doi:10.1214/ECP.v14-1454

Mathematical Reviews number (MathSciNet)
MR2497323

Zentralblatt MATH identifier
1189.60155

Subjects
Primary: 60J80: Branching processes (Galton-Watson, birth-and-death, etc.)
Secondary: 28A78: Hausdorff and packing measures

Keywords
Galton-Watson tree exact Hausdorff measure boundary branching measure size-biased tree

Rights
This work is licensed under aCreative Commons Attribution 3.0 License.

Citation

Duquesne, Thomas. An elementary proof of Hawkes's conjecture on Galton-Watson trees. Electron. Commun. Probab. 14 (2009), paper no. 15, 151--164. doi:10.1214/ECP.v14-1454. https://projecteuclid.org/euclid.ecp/1465234724


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References

  • Aldous, David; Pitman, Jim. Tree-valued Markov chains derived from Galton-Watson processes. Ann. Inst. H. Poincaré Probab. Statist. 34 (1998), no. 5, 637–686.
  • Aldous, David. Asymptotic fringe distributions for general families of random trees. Ann. Appl. Probab. 1 (1991), no. 2, 228–266.
  • Athreya, Krishna B.; Ney, Peter E. Branching processes.Die Grundlehren der mathematischen Wissenschaften, Band 196.Springer-Verlag, New York-Heidelberg, 1972. xi+287 pp.
  • Bingham, N. H.; Doney, R. A. Asymptotic properties of supercritical branching processes. I. The Galton-Watson process. Advances in Appl. Probability 6 (1974), 711–731.
  • Duquesne, Thomas. Continuum random trees and branching processes with immigration. Stochastic Process. Appl. 119 (2009), no. 1, 99–129.
  • Edgar, G. A. Centered densities and fractal measures. New York J. Math. 13 (2007), 33–87 (electronic).
  • Grimmett, G. R. Random labelled trees and their branching networks. J. Austral. Math. Soc. Ser. A 30 (1980/81), no. 2, 229–237.
  • Hawkes, John. Trees generated by a simple branching process. J. London Math. Soc. (2) 24 (1981), no. 2, 373–384.
  • Holmes, R. A. A local asymptotic law and the exact Hausdorff measure for a simple branching process. Proc. London Math. Soc. (3) 26 (1973), 577–604.
  • Kesten, H., and Stigum, B. A limit theorem for multidimensional Galton-Watson processes. Ann. Math. Statist. 37 (1966), 1211–1223.
  • Liu, Quansheng. The exact Hausdorff dimension of a branching set. Probab. Theory Related Fields 104 (1996), no. 4, 515–538.
  • Liu, Quansheng. Exact packing measure on a Galton-Watson tree. Stochastic Process. Appl. 85 (2000), no. 1, 19–28.
  • Liu, Quansheng. Local dimensions of the branching measure on a Galton-Watson tree. Ann. Inst. H. Poincaré Probab. Statist. 37 (2001), no. 2, 195–222.
  • Lyons, Russell. Random walks and percolation on trees. Ann. Probab. 18 (1990), no. 3, 931–958.
  • Lyons, Russell; Pemantle, Robin; Peres, Yuval. Conceptual proofs of $L\log L$ criteria for mean behavior of branching processes. Ann. Probab. 23 (1995), no. 3, 1125–1138.
  • Mörters, Peter; Shieh, Narn-Rueih. Thin and thick points for branching measure on a Galton-Watson tree. Statist. Probab. Lett. 58 (2002), no. 1, 13–22.
  • Mörters, Peter; Shieh, Narn-Rueih. On the multifractal spectrum of the branching measure on a Galton-Watson tree. J. Appl. Probab. 41 (2004), no. 4, 1223–1229.
  • Neveu, J. Arbres et processus de Galton\mhy Watson.(French) [Galton-Watson trees and processes] Ann. Inst. H. Poincaré Probab. Statist. 22 (1986), no. 2, 199–207.
  • Rogers, C. A.; Taylor, S. J. Functions continuous and singular with respect to a Hausdorff measure. Mathematika 8 1961 1–31. (24 #A200)
  • Watanabe, Toshiro. Exact packing measure on the boundary of a Galton-Watson tree. J. London Math. Soc. (2) 69 (2004), no. 3, 801–816.
  • Watanabe, Toshiro. Exact Hausdorff measure on the boundary of a Galton-Watson tree. Ann. Probab. 35 (2007), no. 3, 1007–1038.