Dans l'esprit d'un résultat classique concernant les processus de Crump-Mode-Jagers, nous démontrons une loi forte des grands nombres pour des processus de fragmentation. Plus précisément, pour des processus auto-similaires de fragmentation, incluant les processus homogènes, nous prouvons la convergence presque sûre de la mesure empirique associée à la ligne d'arrêt correspondant aux premiers fragments de taille strictement plus petite qu'un η dans (0, 1].
In the spirit of a classical result for Crump-Mode-Jagers processes, we prove a strong law of large numbers for fragmentation processes. Specifically, for self-similar fragmentation processes, including homogenous processes, we prove the almost sure convergence of an empirical measure associated with the stopping line corresponding to first fragments of size strictly smaller than η for 1≥η>0.
@article{AIHPB_2010__46_1_119_0,
author = {Harris, S. C. and Knobloch, R. and Kyprianou, A. E.},
title = {Strong law of large numbers for fragmentation processes},
journal = {Annales de l'I.H.P. Probabilit\'es et statistiques},
volume = {46},
year = {2010},
pages = {119-134},
doi = {10.1214/09-AIHP311},
mrnumber = {2641773},
zbl = {1195.60046},
language = {en},
url = {http://dml.mathdoc.fr/item/AIHPB_2010__46_1_119_0}
}
Harris, S. C.; Knobloch, R.; Kyprianou, A. E. Strong law of large numbers for fragmentation processes. Annales de l'I.H.P. Probabilités et statistiques, Tome 46 (2010) pp. 119-134. doi : 10.1214/09-AIHP311. http://gdmltest.u-ga.fr/item/AIHPB_2010__46_1_119_0/
[1] and . Strong limit theorems for general supercritical branching processes with applications to branching diffusions. Z. Wahrsch. Verw. Gebiete 36 (1976) 195-212. | MR 420889 | Zbl 0325.60081
[2] and . Strong limit theorems for supercritical immigration-branching processes. Math. Scand. 39 (1977) 327-342. | MR 438498 | Zbl 0348.60117
[3] . Fragmentation of ordered partitions and intervals. Electron J. Probab. 11 (2006) 394-417. | MR 2223041 | Zbl 1109.60058
[4] . Multifractal spectrum of fragmentations. J. Stat. Phys. 113 (2003) 411-430. | MR 2013691 | Zbl 1033.82012
[5] . Subordinators: Examples and applications. In Lectures on Probability Theory and Statistics (Saint-Flour, 1997) 1-91. Lecture Notes in Math. 1717. Springer, Berlin, 1999. | MR 1746300 | Zbl 0955.60046
[6] . Self-similar fragmentations. Ann. Inst. H. Poincaré Probab. Statist. 38 (2002) 319-340. | Numdam | MR 1899456 | Zbl 1002.60072
[7] . The asymptotic behaviour of fragmentation processes. J. Eur. Math. Soc. 5 (2003) 395-416. | MR 2017852 | Zbl 1042.60042
[8] . On small masses in self-similar fragmentations. Stochastic Process. Appl. 109 (2004) 13-22. | MR 2024841 | Zbl 1075.60092
[9] . Random Fragmentation and Coagulation Processes. Cambridge Univ. Press, 2006. | MR 2253162 | Zbl 1107.60002
[10] and . Asymptotic laws for nonconservative self-similar fragmentations. Electron. J. Probab. 9 (2004) 575-593. | MR 2080610 | Zbl 1064.60075
[11] and . Fragmentation energy. Adv. Appl. Probab. 37 (2005) 553-570. | MR 2144567 | Zbl 1080.60080
[12] and . Discritization methods for homogenous fragmentations. J. London Math. Soc. 72 (2005) 91-109. | MR 2145730 | Zbl 1077.60053
[13] and . Additive martingales and probability tilting for homogeneous fragmentations. Unpublished manuscript, 2005. Available at http://www.proba.jussieu.fr/mathdoc/textes/PMA-808.pdf.
[14] and . Asymptotic behaviour of the presence probability in branching random walks and fragmentations. Unpublished manuscript, 2004. Available at http://hal.ccsd.cnrs.fr/ccsd-00002955.
[15] . Martingale convergence in the branching random walk. J. Appl. Probab. 14 (1977) 25-37. | MR 433619 | Zbl 0356.60053
[16] . Uniform convergence of martingales in the branching random walk. Ann. Probab. 20 (1992) 137-151. | MR 1143415 | Zbl 0748.60080
[17] and . Asymptotic properties of supercritical branching processes. II: Crump-Mode and Jirana processes. Adv. Appl. Probab. 7 (1975) 66-82. | MR 378125 | Zbl 0308.60049
[18] and . Limit theorems for branching Markov processes. J. Funct. Anal. 250 (2007) 374-399. | MR 2352485 | Zbl 1125.60087
[19] , and . An almost sure scaling limit theorem for Dawson-Watanabe superprocesses J. Funct. Anal. 254 (2008) 1988-2019. | MR 2397881 | Zbl 1138.60054
[20] . Law of large numbers for superdiffusions: The non-ergodic case. Ann. Inst. H. Poincaré Probab. Statist. (2009). To appear. | Numdam | MR 2500226 | Zbl 1172.60022
[21] , and . Strong law of large numbers for branching diffusion. Ann. Inst. H. Poincaré Probab. Statist. (2009). To appear. | Numdam | MR 2641779 | Zbl 1196.60139
[22] and . Law of large numbers for a class of superdiffusions. Ann. Inst. H. Poincaré Probab. Statist. 42 (2007) 171-185. | Numdam | MR 2199796 | Zbl 1093.60058
[23] and . A spine approach to branching diffusions with applications to Lp-convergence of martingales. In Séminaire de Probabilités, XLII, 2008. To appear, 2009. | MR 2599214 | Zbl 1193.60100
[24] . Convergence of a Gibbs-Boltzmann random measure for a typed branching diffusion. In Séminaire de Probabilités, XXXIV 239-256. Lecture Notes in Math. 1729. Springer, Berlin, 2000. | Numdam | MR 1768067 | Zbl 0985.60053
[25] and . Large-deviations and martingales for a typed branching diffusion: I. Astérisque 236 (1996) 133-154. | MR 1417979 | Zbl 0857.60088
[26] . Branching Processes with Biological Applications. Wiley, London, 1975. | MR 488341 | Zbl 0356.60039
[27] . Multifractal spectra and precise rates of decay in homogeneous fragmentations. Stochastic Process. Appl. 118 (2008) 897-916. | MR 2418249 | Zbl 1141.60064
[28] . A simple path to Biggins' martingale convergence for branching random walk. In Classical and Modern Branching Processes (Minneapolis, MN, 1994) 217-221. IMA Vol. Math. Appl. 84. Springer, New York, 1997. | MR 1601749 | Zbl 0897.60086
[29] . On the convergence of subcritical general (C-M-J) branching processes. Z. Wahrsch. Verw. Gebiete 57 (1981) 365-395. | MR 629532 | Zbl 0451.60078
[30] . Spatial growth of branching processes of particles living in ℝd. Ann. Probab. 10 (1982) 896-918. | MR 672291 | Zbl 0499.60088