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

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.

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].

DOI : https://doi.org/10.1214/09-AIHP311
Classification:  60J25,  60G09
Keywords: fragmentation processes, strong law of large numbers, additive martingales
@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},
     publisher = {Gauthier-Villars},
     volume = {46},
     number = {1},
     year = {2010},
     pages = {119-134},
     doi = {10.1214/09-AIHP311},
     zbl = {1195.60046},
     mrnumber = {2641773},
     language = {en},
     url = {http://www.numdam.org/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, Volume 46 (2010) no. 1, pp. 119-134. doi : 10.1214/09-AIHP311. http://www.numdam.org/item/AIHPB_2010__46_1_119_0/

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