Sur une équation fonctionnelle et ses applications: Une extension du théorème de kesten-stigum concernant des processus de branchement

Advances in Applied Probability

Volumn 29, Issue 2, 1997, Pages 353-373

  Liu, Quansheng ^a  

^a UNIVERSITÉ DE RENNES 1   (France)

Author keywords

Branching processes; Branching random walks; Functional equation; Mandelbrot's martingale

Indexed keywords

EID: 3342908453     PISSN: 00018678     EISSN: None     Source Type: Journal    
DOI: 10.1017/S0001867800028032     Document Type: Article

References (42)

1. ASMUSSEN, S. ET HERING, H. (1983) Branching Processes. Birkhäuser, Basel.
2. ATHREYA, K. B. (1970) A simple proof of a result of Kesten and Stigum on supercritical multitype Galton-Watson branching processes. Ann. Math. Statist. 41, 195-202.
3. ATHREYA, K. B. (1971) A note on a functional equation arising in Galton-Watson branching processes. J. Appl. Prob. 8, 589-598.
4. ATHREYA, K. B. ET NEY, P. E. (1972) Branching Processes. Springer, Berlin.
5. ATHREYA, K. B. ET KAPLAN, N. (1976) Convergence of the age distribution in the one-dimensional supercritical age-independent branching processes. Ann. Prob. 4, 38-50.
6. BEN NASR, F. (1987) Mesures aléatoires de Mandelbrot associées à des substitutions. C.R. Acad. Sci. Paris 304, 255-258.
7. BIGGINS, J. D. (1977) Martingale convergence in the branching random walk. J. Appl. Prob. 14, 25-37.
8. BIGGINS, J. D. (1979) Growth rates in the branching random walk. Z. Wahrscheinlichkeitsth. 48, 17-34.
9. BIGGINS, J. D. (1992) Uniform convergence of martingales in the branching random walk. Ann. Prob. 20, 137-151.
10. BINGHAM, N. H. ET DONEY, R. A. (1974) Asymptotic properties of supercritical branching processes I: the Galton-Watson process. Adv. Appl. Prob. 6, 711-731.
11. BINGHAM, N. H. ET DONEY, R. A. (1975) Asymptotic properties of supercritical branching processes II: Crump-Mode and Jirina processes. Adv. Appl. Prob. 7, 66-82.
12. CHAUVIN, B. ET ROUAULT, A. (1996) Boltzmann-Gibbs weights in the branching random walk. In Classical and Modern Branching Processes. (IMA Volumes in Mathematics and its Applications 84). ed. K. B. Athreya and P. Jagers. Springer, New York. pp. 41-50.
13. COLLET, P. ET KOUKIOU, F. (1992) Large deviations for multiplicative chaos. Commun. Math. Phys. 147, 329-342.
14. CRUMP, K. ET MODE, C. J. (1968) A general age-dependent branching processes (I). J. Math. Anal. Appl. 24, 497-508.
15. CRUMP, K. ET MODE, C. J. (1969) A general age-dependent branching processes (II). J. Math. Anal. Appl. 25, 8-17.
16. DONEY, R. A. (1972) A limit theorem for a class of supercritical branching processes. J. Appl. Prob. 9, 707-724.
17. DONEY, R. A. (1973) On a functional equation for general branching processes. J. Appl. Prob. 10, 198-205.
18. DURRETT, R. ET LIGGETT, TH. (1983) Fixed points of the smoothing transformation. Z. Wahrscheinlichkeitsth. 64, 275-301.
19. FALCONER, K. J. (1986) Random fractals. Math. Proc. Camb. Phil. Soc. 100, 559-582.
20. FALCONER, K. J. (1987) Cut-set sums and tree processes. Math. Proc. Amer. Math. Soc. 101, 337-346.
21. FALCONER, K. J. (1994) The multifractal spectrum of statistically self-similar measures. J. Theoret. Prob. 7, 681-702.
22. FORTET, R. ET MOURIER, E. (1953) Ann. Scient. Ecole Normale Sup. 70, 266-285.
23. FRANCHI, J. (1993) Chaos multiplicatif: Un traitement simple et complet de la fonction de partition. Prépublication 148. Laboratoire de Probabilités de l'Université Paris VI.
24. GRAF, S., MAULDIN, R. D. ET WILLIAMS, S. C. (1988) The exact Hausdorff dimension in random recursive constructions. Mem. Amer. Math. Soc. 71, 381.
25. GUWARC'H, Y. (1990) Sur une extension de la notion de loi semi-stable. Ann. Inst. Henri Poincaré 26, 261-285.
26. HARRIS, T. E. (1948) Branching processes. Ann. Math. Statist. 19, 474-494.
27. HOLLEY, R. AND LIGGETT, TH. (1981) Generalized potlatch and smoothing processes. Z. Wahrscheinlichkeitsth. 55, 165-195.
28. HOLLEY, R, AND WAYMIRE, E. C. (1992) Multifractal dimensions and scaling exponents for strongly bounded random cascades. Ann. Appl. Prob. 2, 819-845.
29. KAHANE, J. P. (1987) Multiplications aléatoires et dimensions de Hausdorff. Ann. Inst. Henri Poincaré 23, 289-296.
30. KAHANE, J. P. ET PEYRIERE, J. (1976) Sur certaines martingales de Benoit Mandelbrot. Adv. Math. 22, 131-345.
31. 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.
32. LIU, Q. (1998) Fixed points of a generalized smoothing transformation and applications to the branching random walk. Adv. Appl. Prob. 30 to appear.
33. MANDELBROT, B. (1974a) Multiplications aléatoires et distributions invariantes par moyenne pondérée aléatoire. C.R. Acad. Sci. Paris 278, 289-292.
34. MANDELBROT, B. (1974b) Multiplications aléatoires et distributions invariantes par moyenne pondérée aléatoire: quelques extensions. C.R. Acad. Sci. Paris 278, 355-358.
35. MAULDIN, R. D. AND WILLIAMS, S. C. (1986) Random constructions, asymptotic geometric and topological properties. Trans. Amer. Math. Soc. 295, 325-346.
36. NEVEU, J. (1972) Martingales à Temps Discret. Masson, Paris.
37. PEYRIERE, J. (1977) Calculs de dimensions de Hausdorff. Duke Math. J. 44, 591-601.
38. ROYER, G. (1984) Distance de Fortet-Mourier et fonctions log-concaves. Ann. Sci. de Clermont-Ferrand 78.
39. WAYMIRE, E. C. ET WILLIAMS, S. C. (1994) A general decomposition theory for random cascades. Bull. Amer. Math. Soc. 31, 216-222.
40. WAYMIRE, E. C. ET WILLIAMS, S. C. (1995) Multiplicative cascades: dimension spectra and dependence. J. Fourier Anal. Appl. (Special issue in honour of J.-P. Kahane.) pp. 489-609.
41. WAYMIRE, E. C. ET WILLIAMS, S. C. (1996) A cascade decomposition theory with applications to Markov and exchangeable cascades. Trans. Amer. Math. Soc. 348, 585-632.
42. WAYMIRE, E. C. ET WILLIAMS, S. C. (1996) Markov cascades. In Classical and Modern Branching Processes. (IMA Volumes in Mathematics and its Applications 84). éd. K. B. Athreya and P. Jagers. Springer, New York. pp. 305-321.