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On Two Measures Defined on the Boundary of a Branching Tree

  • Quansheng Liu
  • Alain Rouault
Chapter
Part of the The IMA Volumes in Mathematics and its Applications book series (IMA, volume 84)

Abstract

Replying to a question of A. Joffe, we show that two random measures defined on the boundary of a Galton-Watson tree are mutually singular. We compare them in a precise way, and we extend this result to marked trees in the framework of random fractals.

AMS(MOS) subject classifications. Primary- 60J80; Secondary: 28A78, 28A80, 05CO5.

Key words and phrases

Galton-Watson tree branching processes Hausdorff dimension random measures 

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References

  1. [1]
    Athreya, K.B. and Ney, P.E. (1972) Branching Processes, Springer Verlag, Berlin.zbMATHCrossRefGoogle Scholar
  2. [2]
    Biggins, J.D. (1977) Martingale convergence in the branching random walk, J. Appl. Probab. 14 25–37.MathSciNetzbMATHCrossRefGoogle Scholar
  3. [3]
    Biggins, J.D. and Bingham, N.H. (1993) Large deviations in the supercritical branching process, Adv. Appl. Prob.25 757–772.MathSciNetzbMATHCrossRefGoogle Scholar
  4. [4]
    Billingsley, P. (1965) Ergodic Theory and Information, Wiley, New York.Google Scholar
  5. [5]
    Dacunha-Castelle, D. et Duflo, M. (1983) Probabilités et Statistiques, 2. Probl¨¨mes ¨¤ temps mobile, Masson, Paris.Google Scholar
  6. [6]
    Dubuc, S. (1971) La densité de la loi limite d’un processus en cascade expansif, Z. Warscheinlichkeitstheorie 19, 281–290.MathSciNetCrossRefGoogle Scholar
  7. [7]
    Falconer, K.J. (1986) Random Fractals, Math. Proc. Camb. Phil. Soc. 100, 559–582.MathSciNetzbMATHCrossRefGoogle Scholar
  8. [8]
    Graf, S., Mauldin, R.D. and Williams, S.C. (1988) The exact Hausdorff dimension in random recursive constructions, Mem. Amer. Math. Soc., 71 n¡ã381.MathSciNetGoogle Scholar
  9. [9]
    Hawkes, J. (1981) Trees generated by a simple branching process. J. London Math. Soc. (2)24 373–384.MathSciNetCrossRefGoogle Scholar
  10. [10]
    Joffe, A. (1978) Remarks on the structure of trees with applications to supercritical Galton-Watson processes. In Advances in Prob. 5, ed. A.Joffe and P.Ney, Dekker, New- York p.263–268.MathSciNetGoogle Scholar
  11. [11]
    Kahane, J.P. et Peyri¨¨re, J. (1976) Sur certaines martingales de Mandelbrot. Adv. in Math., 22,131–145.MathSciNetzbMATHCrossRefGoogle Scholar
  12. [12]
    Liu, Q. (1992) The exact Hausdorff dimensions of a branching set. Prépublication n¡ã135, Laboratoire de Probabilités, Université Paris VI. Google Scholar
  13. [13]
    Liu, Q. (1993) Sur quelques probl¨¨mes ¨¤ propos des processus de branchement, des flots dans les réseaux et des mesures de Hausdorff associées. Th¨¨se, Université Paris VI, Laboratoire de Probabilités. Google Scholar
  14. [14]
    Liu, Q. (1994) Sur une équation fonctionnelle et ses applications (I): une extension du théor¨¨me de Kesten-Stigum concernant des processus de branchement. Prépublicaiion de l’IRMAR. Google Scholar
  15. [15]
    Lyons, R. (1994) A simple path to Biggins’ martingale convergence for branching random walk. preprint Google Scholar
  16. [16]
    Mauldin, R.D. and Williams, S.C. (1986) Random constructions, asymptotic geometric and topological properties, Trans. Amer. Math. Soc. 29 5, 325–346.MathSciNetCrossRefGoogle Scholar
  17. [17]
    Neveu, J. (1986) Arbres et Processus de Galton-Watson. Ann. IHP 22. Google Scholar
  18. [18]
    O’Brien, G.L. (1980) A limit theorem for sample maxima and heavy branches in Galton-Watson trees. J. Appl. Prob. 17 539–545.MathSciNetzbMATHCrossRefGoogle Scholar
  19. [19]
    Peres, Y., Lyons, R., and Pemantle, R. (1995). Ergodic theory on Galton-Watson trees: speed of random walk and dimension of harmonic measure. Erg odic Theory Dynamical Systems 15 593–620.MathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media New York 1997

Authors and Affiliations

  • Quansheng Liu
    • 1
  • Alain Rouault
    • 2
  1. 1.Universite Rennes I, Irmar, Campus de BeaulieuRenne S CedexFrance
  2. 2.Département de MathématiquesUniversite Versailles-Saint QuentinVersailles CedexFrance

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