Differentiability of multiplicative processes related to branching random walks
Barral, Julien
Annales de l'I.H.P. Probabilités et statistiques, Tome 36 (2000), p. 407-417 / Harvested from Numdam
Publié le : 2000-01-01
@article{AIHPB_2000__36_4_407_0,
     author = {Barral, Julien},
     title = {Differentiability of multiplicative processes related to branching random walks},
     journal = {Annales de l'I.H.P. Probabilit\'es et statistiques},
     volume = {36},
     year = {2000},
     pages = {407-417},
     mrnumber = {1785388},
     zbl = {0977.60077},
     language = {en},
     url = {http://dml.mathdoc.fr/item/AIHPB_2000__36_4_407_0}
}
Barral, Julien. Differentiability of multiplicative processes related to branching random walks. Annales de l'I.H.P. Probabilités et statistiques, Tome 36 (2000) pp. 407-417. http://gdmltest.u-ga.fr/item/AIHPB_2000__36_4_407_0/

[1] J.D. Biggins, Martingale convergence in the branching random walk, J. Appl. Probab. 14 (1977) 25-37. | MR 433619 | Zbl 0356.60053

[2] J.D. Biggins, Growth rates in the branching random walk, Z. Wahrsch. Verw. Gebiete 48 (1979) 17-34. | MR 533003 | Zbl 0387.60092

[3] J.D. Biggins, Uniform convergence of martingales in the one-dimensional branching random walk, in: I.V. Basawa, R.L. Taylor (Eds.), Selected Proceedings of the Sheffield Symposium on Applied Probability, 1989, IMS Lectures Notes- Monograph Series, Vol. 18,1991, pp. 159-173. | MR 1193068 | Zbl 0770.60077

[4] J.D. Biggins, Uniform convergence of martingales in the branching random walk, Ann. Probab. 20 (1992) 137-151. | MR 1143415 | Zbl 0748.60080

[5] R. Durrett, Th. Liggett, Fixed points of the smoothing transformation, Z. Wahrsch. Verw. Gebiete 64 (1983) 275-301. | MR 716487 | Zbl 0506.60097

[6] A. Joffe, L. Le Cam, J. Neveu, Sur la loi des grands nombres pour les variables aléatoires de Bernouilli attachées à un arbre dyadique, C. R. Acad. Sci. Paris 278 (1973) 963-964. | MR 341570 | Zbl 0292.60078

[7] J.-P. Kahane, J. Peyrière, Sur certaines martingales de Benoit Mandelbrot, Adv. Math. 22 (1976) 131-145. | MR 431355 | Zbl 0349.60051

[8] J.F.C. Kingman, The first birth problem for an age-dependent branching process, Ann. Probab. 3 (1975) 790-801. | MR 400438 | Zbl 0325.60079

[9] Q. Liu, Sur une équation fonctionnelle et ses applications: une extension du théorème de Kesten-Stigum concernant des processus de branchement, Adv. Appl. Probab. 29 (2) (1997) 353-373. | MR 1450934 | Zbl 0901.60055

[10] Q. Liu, Self-similar cascades and the branching random walk, Preprint, Univ. Rennes 1, 1997.

[11] R. Lyons, A simple path to Biggins' martingale convergence, in: K.B. Athreya, P. Jagers (Eds.), Classical and Modem Branching Processes, IMA Volumes in Mathematics and its Applications, Vol. 84, Springer, Berlin, 1997, pp. 217-222. | MR 1601749 | Zbl 0897.60086

[12] B. Mandelbrot, Intermittent turbulence in self-similar cascades, divergence of hight moments and dimension of the carrier, J. Fluid. Mech. 62 (1974) 331-358. | Zbl 0289.76031

[13] B. Mandelbrot, Multiplications aléatoires itérées et distributions invariantes par moyennes pondérées, C. R. Acad. Sci. Paris 278 (1974) 289-292, 355-358. | Zbl 0276.60096

[14] J. Neveu, Martingales à Temps Discret, Masson et Cie, Paris, 1972. | MR 402914

[15] E.M. Stein, A. Zygmund, Smoothness and differentiability of functions, Annales Univ. Sci. Budapest. III-IV (1960-1961) 295-307. | MR 132135 | Zbl 0104.04803

[16] N.N. Tchentov, 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. 1 (1956) 140-144.

[17] B. Von Bahr, C.-G. Esseen, Inequalities for the rth absolute moment of sum of random variables, 1 ≤ r ≤ 2, Ann. Math. Statist. 36 (1965) 299-303. | MR 170407 | Zbl 0134.36902

[18] S. Watanabe, Limit theorem for a class of branching processes, in: J. Chover (Ed.), Markov Processes and Potential Theory, Wiley, New York, 1967, pp. 205-232. | MR 237007 | Zbl 0253.60072