no. 1
Strong law of large numbers for fragmentation processes
Annales de l'I.H.P. Probabilités et statistiques, Tome 46 (2010) no. 1, pp. 119-134.

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.

DOI : 10.1214/09-AIHP311
Classification : 60J25, 60G09
Mots clés : fragmentation processes, strong law of large numbers, additive martingales 
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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) no. 1, pp. 119-134. doi : 10.1214/09-AIHP311. http://archive.numdam.org/articles/10.1214/09-AIHP311/

[1] S. Asmussen and H. Hering. Strong limit theorems for general supercritical branching processes with applications to branching diffusions. Z. Wahrsch. Verw. Gebiete 36 (1976) 195-212. | MR | Zbl

[2] S. Asmussen and H. Hering. Strong limit theorems for supercritical immigration-branching processes. Math. Scand. 39 (1977) 327-342. | MR | Zbl

[3] A.-L. Basdevant. Fragmentation of ordered partitions and intervals. Electron J. Probab. 11 (2006) 394-417. | MR | Zbl

[4] J. Berestycki. Multifractal spectrum of fragmentations. J. Stat. Phys. 113 (2003) 411-430. | MR | Zbl

[5] J. Bertoin. 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 | Zbl

[6] J. Bertoin. Self-similar fragmentations. Ann. Inst. H. Poincaré Probab. Statist. 38 (2002) 319-340. | Numdam | MR | Zbl

[7] J. Bertoin. The asymptotic behaviour of fragmentation processes. J. Eur. Math. Soc. 5 (2003) 395-416. | MR | Zbl

[8] J. Bertoin. On small masses in self-similar fragmentations. Stochastic Process. Appl. 109 (2004) 13-22. | MR | Zbl

[9] J. Bertoin. Random Fragmentation and Coagulation Processes. Cambridge Univ. Press, 2006. | MR | Zbl

[10] J. Bertoin and A. Gnedin. Asymptotic laws for nonconservative self-similar fragmentations. Electron. J. Probab. 9 (2004) 575-593. | MR | Zbl

[11] J. Bertoin and S. Martinez. Fragmentation energy. Adv. Appl. Probab. 37 (2005) 553-570. | MR | Zbl

[12] J. Bertoin and A. Rouault. Discritization methods for homogenous fragmentations. J. London Math. Soc. 72 (2005) 91-109. | MR | Zbl

[13] J. Bertoin and A. Rouault. Additive martingales and probability tilting for homogeneous fragmentations. Unpublished manuscript, 2005. Available at http://www.proba.jussieu.fr/mathdoc/textes/PMA-808.pdf.

[14] J. Bertoin and A. Rouault. 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] J. D. Biggins. Martingale convergence in the branching random walk. J. Appl. Probab. 14 (1977) 25-37. | MR | Zbl

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

[17] N. Bingham and R. A. Doney. Asymptotic properties of supercritical branching processes. II: Crump-Mode and Jirana processes. Adv. Appl. Probab. 7 (1975) 66-82. | MR | Zbl

[18] Z.-Q. Chen and Y. Shiozawa. Limit theorems for branching Markov processes. J. Funct. Anal. 250 (2007) 374-399. | MR | Zbl

[19] Z.-Q. Chen, Y. Ren and H. Wang. An almost sure scaling limit theorem for Dawson-Watanabe superprocesses J. Funct. Anal. 254 (2008) 1988-2019. | MR | Zbl

[20] J. Engländer. Law of large numbers for superdiffusions: The non-ergodic case. Ann. Inst. H. Poincaré Probab. Statist. (2009). To appear. | Numdam | MR | Zbl

[21] J. Engländer, S. C. Harris and A. E. Kyprianou. Strong law of large numbers for branching diffusion. Ann. Inst. H. Poincaré Probab. Statist. (2009). To appear. | Numdam | MR | Zbl

[22] J. Engländer and A. Winter. Law of large numbers for a class of superdiffusions. Ann. Inst. H. Poincaré Probab. Statist. 42 (2007) 171-185. | Numdam | MR | Zbl

[23] R. Hardy and S. C. Harris. A spine approach to branching diffusions with applications to Lp-convergence of martingales. In Séminaire de Probabilités, XLII, 2008. To appear, 2009. | MR | Zbl

[24] S. C. Harris. 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 | Zbl

[25] S. C. Harris and D. Williams. Large-deviations and martingales for a typed branching diffusion: I. Astérisque 236 (1996) 133-154. | Numdam | MR | Zbl

[26] P. Jagers. Branching Processes with Biological Applications. Wiley, London, 1975. | MR | Zbl

[27] N. Krell. Multifractal spectra and precise rates of decay in homogeneous fragmentations. Stochastic Process. Appl. 118 (2008) 897-916. | MR | Zbl

[28] R. Lyons. 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 | Zbl

[29] O. Nerman. On the convergence of subcritical general (C-M-J) branching processes. Z. Wahrsch. Verw. Gebiete 57 (1981) 365-395. | MR | Zbl

[30] K. Uchiyama. Spatial growth of branching processes of particles living in ℝd. Ann. Probab. 10 (1982) 896-918. | MR | Zbl

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