Fixed Points of a Generalized Smoothing Transformation and Applications to Branching Processes

Quansheng Liu

Publications mathématiques et informatique de Rennes (1995)

  • Issue: 2, page 1-54

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Liu, Quansheng. "Fixed Points of a Generalized Smoothing Transformation and Applications to Branching Processes." Publications mathématiques et informatique de Rennes (1995): 1-54. <http://eudml.org/doc/274601>.

@article{Liu1995,
author = {Liu, Quansheng},
journal = {Publications mathématiques et informatique de Rennes},
language = {eng},
number = {2},
pages = {1-54},
publisher = {Département de Mathématiques et Informatique, Université de Rennes},
title = {Fixed Points of a Generalized Smoothing Transformation and Applications to Branching Processes},
url = {http://eudml.org/doc/274601},
year = {1995},
}

TY - JOUR
AU - Liu, Quansheng
TI - Fixed Points of a Generalized Smoothing Transformation and Applications to Branching Processes
JO - Publications mathématiques et informatique de Rennes
PY - 1995
PB - Département de Mathématiques et Informatique, Université de Rennes
IS - 2
SP - 1
EP - 54
LA - eng
UR - http://eudml.org/doc/274601
ER -

References

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  1. Asmussen, S. and Hering, H. (1983) : Branching processes. Birkhäuser. Zbl0516.60095MR701538
  2. Athreya, K.B. (1971) : A note on a functional equation arising in Galton-Watson branching processes. J. Appl. Prob., 8, 589-598. Zbl0254.60058MR292184
  3. Athreya, K.B. and Ney, P.E. (1972) : Branching processes. Springer, Berlin. Zbl0259.60002MR373040
  4. Ben Nasr, F. (1987) : Mesures aléatoires de Mandelbrot associées a des substitutions. C.R. Acad. Sci.Paris, 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. Bingham, N.H. and Doney, R.A. (1974) : Asympototic properties of supercritical branching processes I : The Galton-Watson process. Adv. Appl. Prob., 6, 711-731. Zbl0297.60044MR362525
  7. Bingham, N.H. and Doney, R.A. (1975): Asympototic properties of supercritical branching processes II: Crump-Mode and Jirina processes. Adv. Appl. Prob., 7, 66-82. Zbl0308.60049MR378125
  8. Chauvin, B. and Rouault, A. (1993) : Boltzmann-Gibbs weights in the branching random walk. Prépublication no.175 du Laboratoire de Probabilités de l'Université Paris VI. Zbl0866.60074
  9. Collet, P. and Koukiou, F. (1992) : Large deviations for multiplicative chaos. Commun. Math. Phys.147, 329-342. Zbl0755.60022MR1174416
  10. Crump, K. and Mode, C.J. (1968) : A general age-dependent branching process (I). J. Math. Anal. Appl., 24,497-508. Zbl0192.54301MR237005
  11. Crump, K. and Mode, C.J. (1969) : A general age-dependent branching process (II). J. Math. Anal. Appl., 25, 8-17. Zbl0201.19202MR237005
  12. Doney, R.A. (1972) : A limit theorem for a class of supercritical branching processes. J. Appl. Prob., 9, 707-724. Zbl0267.60082MR348853
  13. Doney, R.A. (1973) : On a functional equation for general branching processes. J. Appl. Prob., 10, 497-508. Zbl0258.60060MR350883
  14. Durrett, R. and Liggett, T. (1983) : Fixed points of the smoothing transformation. Z. Wahrscheinlichkeitstheorie verw. Gebiete, 64, 275-301. Zbl0506.60097MR716487
  15. Falconer, K.J. (1986) : Random fractals. Math.Proc.Camb.Phil.Soc., 100, 559-582. Zbl0623.60020MR857731
  16. Falconer, K.J. (1987) : Cut-set processes and tree processes. Math. Proc. Amer. Math. Soc. (2), 101, 337-346. Zbl0636.90031MR902553
  17. Feller, W. (1971) : An introduction to probability theory and its applications. Vol.2, 2nd ed. John Wiley & Sons. Zbl0039.13201
  18. Fortet, R. et Mourier, E (1953) : Convergence de la répartition empirique vers la répartion théorique. Ann. Scient. Ecole Normale Sup. Série 3, tome 70, 266-285. Zbl0053.09601MR61325
  19. Franchi, J. (1993) : Chaos multiplicatif : un traitement simple et complet de la fonction de la partition. Prépublication no. 148 du Laboratoire de Probabilités de l'Université Paris VI. Zbl0834.60101
  20. Graf, S., Mauldin, R.D. and Williams, S.C. (1988) : The exact Hausdorff dimension in random recursive constructions. Mem. Amer. Math. Soc.71, n°381. Zbl0641.60003MR920961
  21. Guivarc'h, Y. (1990) : Sur une extension de la notion semi-stable, Ann. Inst. Henri Poincaré, (Prob.et Stat.), 26, 261-285. Zbl0703.60012MR1063751
  22. Heyde, C.C. (1970) : Extension of a result of Seneta for the supercritical Galton-Watson process. Ann. Math. Statist.41, 739-742. Zbl0195.19201MR254929
  23. Holley, R. and Liggett, T. (1981) : Generalized potlatch and smoothing processes. Z. Wahrscheinlichkeitstheorie verw. Gebiete, 55, 165-195. Zbl0441.60096MR608015
  24. Holley, R. and Waymire, E.C. (1992) : Multifractal dimensions and scaling exponents for strongly bounded random cascades. Ann. Appl. Prob.2, 819-845. Zbl0786.60064MR1189419
  25. Kahane, J.P. (1987) : Multiplications aléatoires et dimensions de Hausdorff. Ann. Inst. Henri Poincaré, Sup.au no.2, vol.23, p.289-296. Zbl0619.60005MR898497
  26. Kahane, J.P. and Peyrière J. (1976) : Sur certaines martingales de Benoit Mandelbrot. Adv. Math., 22, 131-145. Zbl0349.60051MR431355
  27. Kesten, H. and Stigum, B.P. (1966) : A limit theorem for multidimensional Galton-Watson processes, Ann. Math. Statist.37, 1211-1223. Zbl0203.17401MR198552
  28. Liu, Q. (1993) : Sur quelques problèmes à propos des processus de branchement, des flots dans les réseaux et des mesures de Hausdorff asociées. Thèse, Université Paris 6. 
  29. Liu, Q. (1994) : Sur une equation fonctionnelle et ses applications : une extension du théorème de Kesten-Stigum concernant des processus de branchement. Prépublication, Université Rennes 1. Zbl0901.60055
  30. Liu, Q. (1994a) : Growth of entire characteristic functions and tails in tree martingales, Prépublication, Université Rennes 1. 
  31. Mandelbrot, B. (1974) : Multiplications aléatoires et distributions invariantes par moyenne pondérée aléatoire. C.R. Acad. Sci. Paris, vol.278, Série A, 325-346. Zbl0276.60096
  32. Mandelbrot, B. (1974) : Multiplications aléatoires et distributions invariantes par moyenne pondérée aléatoire : Quelques extensions . C.R. Acad. Sci.Paris, vol.278, Série A, 355-358. Zbl0276.60097MR431352
  33. Mauldin, R.A. and Williams, S.C. (1986) : Random constructions, asymptotic geometric and topological properties. Trans. Amer. Math. Soc., 295, 325-346. Zbl0625.54047MR831202
  34. Royer, G. (1984) : Distance de Fortet-Mourier et fonctions log-concaves. Annales Scientifiques de Clermont-Ferrand, n° 78. Zbl0546.60041MR782930
  35. Seneta, E. (1968) : On recent theorems concerning the supercritical Galton-Watson process. Ann. Math. Stat.39, 2098-2102. Zbl0176.47603MR234530
  36. Seneta, E. (1969) : Functional equations and the Galton-Watson process. Adv. Appl. Prob.1, 1-42. Zbl0183.46105MR248917
  37. Seneta, E. (1973) : A Tauberian theorem of E. Landau and W. Feller, Ann. Prob.1, 1057-1058. Zbl0271.60024MR358133
  38. Seneta, E. (1974) : Regular varing functions in the theory of simple branching processes. Adv. Appl. Prob.6, 408-420. Zbl0291.60043MR420894
  39. Waymire, E.C. and Williams, S.C. (1993) : Multiplicative cascades : spectra and dependence. Preprint, Origon State University and Utah State University. Zbl0889.60050

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