Martingales and Rates of Presence in Homogeneous Fragmentations

Nathalie Krell (1) , Alain Rouault (2)

1 MAP5 - UMR 8145 - Mathématiques Appliquées Paris 5
2 LMV - Laboratoire de Mathématiques de Versailles

In mass-conservative homogeneous fragmentations, sizes of the fragments decrease at {\bf asymptotic} exponential rates. Like in branching processes, two situations occur: either the number of such fragments is exponentially growing - the rate is effective -, or the probability of presence of such fragments is exponentially decreasing. In a recent paper, N. Krell considers fragments whose sizes decrease at {\bf exact} exponential rates. In this new setting, she characterizes the effective rates and studies Hausdorff dimension. The present paper carries out a detailed analysis of this model and focus on presence probabilities, using the spine method and a suitable martingale. For the sake of completeness, we compare our results with results and methods of the classical model.

Mots clés: fragmentation; Lévy process; martingales; probability tilting

Domaines: Probabilités [math.PR]

Nathalie Krell, Alain Rouault. Martingales and Rates of Presence in Homogeneous Fragmentations. 2009. ⟨hal-00374031v1⟩