Behavior near the extinction time in self-similar fragmentations I : the stable case

Goldschmidt, Christina; Haas, Bénédicte

Annales de l'I.H.P. Probabilités et statistiques, Tome 46 (2010) no. 2, pp. 338-368.

La fragmentation stable d'incice α∈[-1/2, 0) est construite à partir des masses des sous-arbres de l'arbre continu aléatoire stable d'indice (1+α)-1 obtenus en ne gardant que les feuilles situées à une hauteur supérieure à t, pour t≥0. Nous donnons une description détaillée du comportement asymptotique d'une telle fragmentation, (F(t), t≥0), au voisinage de son point d'extinction, ζ. En particulier, nous montrons que t1/αF((ζ-t)+) converge en loi lorsque t→0 vers une limite non triviale. Pour obtenir ce résultat, nous allons plus loin et décrivons le comportement asymptotique en loi, après normalisation, (a) d'une excursion du processus de hauteur stable (conditionnée à avoir une longueur 1) au voisinage de son maximum; (b) des intervalles ouverts où l'excursion est au-dessus d'un certain niveau; et (c) de la suite décroissante des longueurs de ces intervalles. Notre outil principal est la théorie des excursions. Nous nous intéressons également au dernier fragment à disparaître et montrons, qu'avec les mêmes normalisations en temps et espace, la masse de ce fragment a une distribution limite construite à partir d'une certaine version biaisée de ζ. Enfin, nous montrons que les logarithmes des masses du plus gros fragment et du dernier fragment à disparaître, au temps (ζ-t)+, divisés par log(t), convergent presque sûrement vers la constante -1/α lorsque t→0.

The stable fragmentation with index of self-similarity α∈[-1/2, 0) is derived by looking at the masses of the subtrees formed by discarding the parts of a (1+α)-1-stable continuum random tree below height t, for t≥0. We give a detailed limiting description of the distribution of such a fragmentation, (F(t), t≥0), as it approaches its time of extinction, ζ. In particular, we show that t1/αF((ζ-t)+) converges in distribution as t→0 to a non-trivial limit. In order to prove this, we go further and describe the limiting behavior of (a) an excursion of the stable height process (conditioned to have length 1) as it approaches its maximum; (b) the collection of open intervals where the excursion is above a certain level; and (c) the ranked sequence of lengths of these intervals. Our principal tool is excursion theory. We also consider the last fragment to disappear and show that, with the same time and space scalings, it has a limiting distribution given in terms of a certain size-biased version of the law of ζ. In addition, we prove that the logarithms of the sizes of the largest fragment and last fragment to disappear, at time (ζ-t)+, rescaled by log(t), converge almost surely to the constant -1/α as t→0.

Classification : 60G18, 60G52, 60J25
Mots clés : stable Lévy processes, height processes, self-similar fragmentations, extinction time, scaling limits

[1] R. Abraham and J.-F. Delmas. Williams' decomposition of the Lévy continuous random tree and simultaneous extinction probability for populations with neutral mutations. Stochastic Process. Appl. 119 (2009) 1124-1143. | MR | Zbl

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

[3] J. Bertoin. Lévy Processes. Cambridge Tracts in Mathematics 121. Cambridge Univ. Press, Cambridge, 1996. | MR | Zbl

[4] 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

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

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

[7] J. Bertoin. Random Fragmentation and Coagulation Processes. Cambridge Studies in Advanced Mathematics 102. Cambridge Univ. Press, Cambridge, 2006. | MR | Zbl

[8] J. Bertoin and A. Rouault. Discretization methods for homogeneous fragmentations. J. London Math. Soc. (2) 72 (2005) 91-109. | MR | Zbl

[9] P. Biane, J. Pitman and M. Yor. Probability laws related to the Jacobi theta and Riemann zeta functions, and Brownian excursions. Bull. Amer. Math. Soc. (N.S.) 38 (2001) 435-465 (electronic). | MR | Zbl

[10] P. Carmona, F. Petit and M. Yor. On the distribution and asymptotic results for exponential functionals of Lévy processes. In Exponential Functionals and Principal Values Related to Brownian Motion 73-130. Bibl. Rev. Mat. Iberoamericana. Rev. Mat. Iberoamericana, Madrid, 1997. | MR | Zbl

[11] T. Duquesne and J.-F. Le Gall. Random trees, Lévy processes and spatial branching processes. Astérisque 281 (2002) 1-147. | Numdam | MR | Zbl

[12] C. Goldschmidt and B. Haas. Behavior near the extinction time in self-similar fragmentations II: Finite dislocation measures. In preparation, 2010.

[13] B. Haas. Loss of mass in deterministic and random fragmentations. Stochastic Process. Appl. 106 (2003) 245-277. | MR | Zbl

[14] B. Haas. Regularity of formation of dust in self-similar fragmentations. Ann. Inst. H. Poincaré Probab. Statist. 40 (2004) 411-438. | Numdam | MR | Zbl

[15] B. Haas. Fragmentation processes with an initial mass converging to infinity. J. Theoret. Probab. 20 (2007) 721-758. | MR | Zbl

[16] B. Haas and G. Miermont. The genealogy of self-similar fragmentations with negative index as a continuum random tree. Electron. J. Probab. 9 (2004) 57-97 (electronic). | MR | Zbl

[17] J. Jacod and A. N. Shiryaev. Limit Theorems for Stochastic Processes, 2nd edition. Grundlehren der Mathematischen Wissenschaften 288. Springer, Berlin, 2003. | MR | Zbl

[18] D. P. Kennedy. The distribution of the maximum Brownian excursion. J. Appl. Probab. 13 (1976) 371-376. | MR | Zbl

[19] J.-F. Le Gall and Y. Le Jan. Branching processes in Lévy processes: The exploration process. Ann. Probab. 26 (1998) 213-252. | MR | Zbl

[20] G. Miermont. Self-similar fragmentations derived from the stable tree. I. Splitting at heights. Probab. Theory Related Fields 127 (2003) 423-454. | MR | Zbl

[21] J. Pitman. Combinatorial Stochastic Processes. Lecture Notes in Math. 1875. Springer, Berlin, 2006. Lectures from the 32nd Saint-Flour Summer School on Probability Theory. | MR | Zbl

[22] L. C. G. Rogers and D. Williams. Diffusions, Markov Processes and Martingales, Vol. 2: Itô Calculus. Cambridge Univ. Press, Cambridge, 2000. | MR | Zbl

[23] G. Uribe Bravo. The falling apart of the tagged fragment and the asymptotic disintegration of the Brownian height fragmentation. Ann. Inst. H. Poincaré Probab. Statist. To appear, 2010. Available at arXiv:0811.4754. | Numdam | MR | Zbl