Differentiability of multiplicative processes related to branching random walks

Annales de l'institut Henri Poincare (B) Probability and Statistics

Volumn 36, Issue 4, 2000, Pages 407-417

  Barral, Julien ^a  

^a UNIVERSITÉ PARIS SACLAY   (France)

Author keywords

Branching random walks; Functional equations; Martingales; Multiplicative cascades

Indexed keywords

EID: 0042686173     PISSN: 02460203     EISSN: None     Source Type: Journal    
DOI: 10.1016/S0246-0203(00)00125-4     Document Type: Article

References (18)

1. Biggins J.D. Martingale convergence in the branching random walk. J. Appl. Probab. Vol. 14:1977;25-37.
2. Biggins J.D. Growth rates in the branching random walk. Z. Wahrsch. Verw. Gebiete. Vol. 48:1979;17-34.
3. Biggins J.D. Uniform convergence of martingales in the one-dimensional branching random walk. Basawa I.V., Taylor R.L. Selected Proceedings of the Sheffield Symposium on Applied Probability, 1989, IMS Lectures Notes - Monograph Series. Vol. 18:1991;159-173.
4. Biggins J.D. Uniform convergence of martingales in the branching random walk. Ann. Probab. Vol. 20:1992;137-151.
5. Durrett R., Liggett Th. Fixed points of the smoothing transformation. Z. Wahrsch. Verw. Gebiete. Vol. 64:1983;275-301.
6. Joffe A., Le Cam L., Neveu J. Sur la loi des grands nombres pour les variables aléatoires de Bernouilli attachées à un arbre dyadique. C. R. Acad. Sci. Paris. Vol. 278:1973;963-964.
7. Kahane J.-P., Peyrière J. Sur certaines martingales de Benoît Mandelbrot. Adv. Math. Vol. 22:1976;131-145.
8. Kingman J.F.C. The first birth problem for an age-dependent branching process. Ann. Probab. Vol. 3:1975;790-801.
9. Liu Q. Sur une équation fonctionnelle et ses applications: une extension du théorème de Kesten-Stigum concernant des processus de branchement. Adv. Appl. Probab. Vol. 29:(2):1997;353-373.
10. Liu Q. Self-similar cascades and the branching random walk. Preprint, Univ. Rennes 1. 1997.
11. Lyons R. A simple path to Biggins' martingale convergence. Athreya K.B., Jagers P. Classical and Modern Branching Processes, IMA Volumes in Mathematics and its Applications. Vol. 84:1997;217-222 Springer, Berlin.
12. Mandelbrot B. Intermittent turbulence in self-similar cascades, divergence of hight moments and dimension of the carrier. J. Fluid. Mech. Vol. 62:1974;331-358.
13. Mandelbrot B. Multiplications aléatoires itérées et distributions invariantes par moyennes pondérées. C. R. Acad. Sci. Paris. Vol. 278:1974;289-292. 355-358.
14. ie . 1972;Paris.
15. Stein E.M., Zygmund A. Smoothness and differentiability of functions. Annales Univ. Sci. Budapest. Vol. III-IV:1960-1961;295-307.
16. Tchentov N.N. Weak convergence of stochastic processes whoose trajectories have no discontinuity of the second kind and the "heuristic" approach to the Kolmogorov-Smirnov tests. Theory Probab. Appl. Vol. 1:1956;140-144.
17. Von Bahr B., Esseen C.-G. Inequalities for the r th absolute moment of sum of random variables, 1≤r≤2 . Ann. Math. Statist. Vol. 36:1965;299-303.
18. Watanabe S. Limit theorem for a class of branching processes. Chover J. Markov Processes and Potential Theory. 1967;205-232 Wiley, New York.