Strong law of large numbers for fragmentation processes

Harris, S. C.; Knobloch, R.; Kyprianou, A. E.

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.

Classification : 60J25, 60G09
Mots clés : fragmentation processes, strong law of large numbers, additive martingales

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