Products of Random Weights Indexed by Galton-Watson Trees

Quansheng Liu

Publications mathématiques et informatique de Rennes (1996-1997)

  • Issue: 2, page 1-24

How to cite


Liu, Quansheng. "Products of Random Weights Indexed by Galton-Watson Trees." Publications mathématiques et informatique de Rennes (1996-1997): 1-24. <>.

author = {Liu, Quansheng},
journal = {Publications mathématiques et informatique de Rennes},
keywords = {self-similar cascades; branching random walk; functional equations; moments; tails; continuity},
language = {eng},
number = {2},
pages = {1-24},
publisher = {Département de Mathématiques et Informatique, Université de Rennes},
title = {Products of Random Weights Indexed by Galton-Watson Trees},
url = {},
year = {1996-1997},

AU - Liu, Quansheng
TI - Products of Random Weights Indexed by Galton-Watson Trees
JO - Publications mathématiques et informatique de Rennes
PY - 1996-1997
PB - Département de Mathématiques et Informatique, Université de Rennes
IS - 2
SP - 1
EP - 24
LA - eng
KW - self-similar cascades; branching random walk; functional equations; moments; tails; continuity
UR -
ER -


  1. Athreya, K.B. (1971) : A note on a functional equation arising in Galton-Watson branching processes. J. Appl. Prob., 8, 589-598. Zbl0254.60058MR292184
  2. Athreya, K.B. and Ney, P.E. (1972) : Branching processes. Springer, Berlin. Zbl0259.60002MR373040
  3. Barrai J. (1997): Continuté, Moments d'ordres négatifs et Analyse Multifractale de cascades multiplicatives de Mandelbrot. Thèse, Univ. Paris-Sud, Orsay. 
  4. Ben Nasr, F. (1987) : Mesures aléatoires de Mandelbrot associées à des substitutions. CRAS, Sér. I, 304, 255-258. Zbl0635.60052MR882783
  5. Biggins, J.D. (1977) : Martingale convergence in the branching random walk. J. Appl. Prob.14, 25-37. Zbl0356.60053MR433619
  6. Biggins, J.D. and Bingham, N.H. (1993) : Large deviations in the supercritical branching process. Adv. Appl. Prob.25, 757-772. Zbl0796.60090MR1241927
  7. Bingham, N.H. and Doney, R.A. (1974): Asymptotic properties of supercritical ranching processes I: The Galton-Watson process. Adv. Appl. Prob., 6, 711-731. Zbl0297.60044MR362525
  8. Bingham, N.H. and Doney, R.A. (1975): Asymptotic properties of supercritical branching processes II: Crump-Mode and Jirina processes. Adv. Appl. Prob., 7, 66-82. Zbl0308.60049MR378125
  9. Bingham, N.H., Goldie, C.M. and Teugels, J.L. (1987) : Regular variation. Cambridge: Cambridge University Press. Zbl0617.26001MR898871
  10. Chauvin, B. and Rouault, A. (1996) Boltzmann-Gibbs weights in the branching random walk. IMA Congress on Branching Processes n°84. Lecture Notes in Maths. Zbl0866.60074MR1601693
  11. Collet, P. and Koukiou, F. (1992) : Large deviations for multiplicative chaos. Comm. Math. Phys.147, 329-342. Zbl0755.60022MR1174416
  12. Crump, K. and Mode, C.J. (1968-1969) : A general age-dependent branching processJ. Math. Anal. Appl., 24, 497-508 and 25, 8-17. Zbl0201.19202MR237005
  13. Doney, R.A. (1972) : A limit theorem for a class of supercritical branching processes. J. Appl. Prob., 9, 707-724. Zbl0267.60082MR348853
  14. Doney, R.A. (1973) : On a functional equation for general branching processes. J. Appl. Prob., 10, 497-508. Zbl0258.60060MR350883
  15. Durrett, R. and Liggett, T. (1983) : Fixed points of the smoothing transformation. Z. Wahrsch. verw. Gebeite, 64, 275-301. Zbl0506.60097MR716487
  16. Falconer, K.J. (1986) : Random fractals. Math.Proc.Camb.Phil.Soc, 100, 559-582. Zbl0623.60020MR857731
  17. Falconer, K.J. (1987) : Cut set sums and tree processes. Proc. Amer. Math. Soc. (2), 101, 337-346. Zbl0636.90031MR902553
  18. Feller, W. (1970): An introduction to probability theory and its applications, vol.11, 2nd. ed. John Wiley & Sons, New York. Zbl0039.13201MR88081
  19. Graf, S. Mauldin, R.D, and Williams, S.C. (1988) : The exact Hausdorff dimension in random recursive constructions. Mem. Amer. Math. Soc.71, 381. Zbl0641.60003MR920961
  20. Grintsevichyus, A.K. (1974): On the continuity of a sum of dependent variables connected with independent walks on lines. Theory Prob. Appl.19, 163-168. Zbl0321.60053MR345178
  21. Grintsevichyus (1975): One limit distribution for a random walk on the line. Lithunian Math. Trans.15, 580-589. Zbl0373.60009
  22. Guivarc'h, Y. (1990) : Sur une extension de la notion de loi semi-stable. Ann. IHP26, 261-285. Zbl0703.60012MR1063751
  23. Harris, T.E. (1948) : Branching processes, Ann. Math. Stat.19, 474-494. Zbl0041.45603MR27465
  24. Holley, R. and Liggett, T. (1981) : Generalized potlach and smoothing processes. Z. Wahrsch. verw. Gebeite, 55, 165-195. Zbl0441.60096MR608015
  25. Holley, R. and Waymire, E.C. (1992) : Multifractal dimensions and scaling exponents for strongly bounded cascades. Ann. Appl. Prob.2, 819-845. Zbl0786.60064MR1189419
  26. Kahane, J.P. (1987) : Multiplications aléatoires et dimension de Hausdorff. Ann. IHP, Sup. au no. 2, vol. 23, 289-296. Zbl0619.60005MR898497
  27. Kahane, J.P. and Peyrière (1976) : Sur certaines martingales de Benoit Mandelbrot. Adv. Math., 22, 131-145. Zbl0349.60051MR431355
  28. Kesten, H. (1973): Random difference equations and renewal theory for products of random matrices. Acta Math.131, 207-248. Zbl0291.60029MR440724
  29. Kesten, H and Stigum, B.P. (1966): A limit theorem for multidimensional Galton- Watson processes. Ann. Math. Statist.37, 1211-1223. Zbl0203.17401MR198552
  30. Liu, Q. (1993) : Sur quelques problèmes à propos des processus de branchement, des flots dans les réseaux et des mesures de Hausdorff associées. Thèse, Université Paris 6. 
  31. Liu, Q. (1996): The exact Hausdorff dimension of a branching set. Prob. Th. Rel. Fields, 104; 1996, 515-538. Zbl0842.60084MR1384044
  32. Liu, Q. (1996): The growth of an entire characteristic function and the tail probabilities of the limit of a tree martingale. In Trees. Progress in Probability, vol.40, pp 51-80, 1996, Birkhöuser: Verlag Basel. Eds.: B.Chauvin, S.Cohen, A.Rouault. Zbl0864.60012MR1439972
  33. Liu, Q. (1997a) : Sur une équation fonctionnelle et ses applications : une extension du théorème de Kesten-Stigum concernant des processus de branchement. To appear in Adv. Appl. Prob.1997. Zbl0901.60055MR1450934
  34. Liu, Q. (1997b) : Fixed points of a generalized smoothing transformation and applications to branching processes. To appear in Adv. Appl. Prob.1997. Zbl0909.60075MR1396815
  35. Liu, Q and Rouault, A. (1996): On two measures defined on the boundary of a branching tree. In "Classical and modern branching processes", ed. K.B. Athreya and P. Jagers. IMA Volumes in Mathematics and its applications, vol. 84, 1996, pp. 187-202. Springer-Verlag. Zbl0867.60065MR1601741
  36. Mandelbrot, B. (1974) : Multiplications aléatoires et distributions invariantes par moyenne pondérée aléatoire. CRASParis vol. 278, 325-346 et 355-358. Zbl0276.60097MR431352
  37. Mauldin, R.A. and Williams, S.C. (1986) : Random constructions, asymptotic geometric and topological properties. Trans. Amer. Math. Soc.295, 325-346. Zbl0625.54047MR831202
  38. Seneta, E. (1968) : On recent theorems concerning the supercritical Galton- Watson process. Ann. Math. Stat.39, 2098-2102. Zbl0176.47603MR234530
  39. Seneta, E. (1969) : Functional equations and the Galton-Watson process. Adv. Appl. Prob.1, 1-42. Zbl0183.46105MR248917
  40. Waymire, E.C. and Williams, S.C. (1995) : Multiplicative cascades : dimension spectra and dependence. J. of Fourier Analysis and Appl., Kahane Special Issue. Zbl0889.60050MR1364911

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