Ranked fragmentations
ESAIM: Probability and Statistics, Tome 6 (2002), pp. 157-175.

In this paper we define and study self-similar ranked fragmentations. We first show that any ranked fragmentation is the image of some partition-valued fragmentation, and that there is in fact a one-to-one correspondence between the laws of these two types of fragmentations. We then give an explicit construction of homogeneous ranked fragmentations in terms of Poisson point processes. Finally we use this construction and classical results on records of Poisson point processes to study the small-time behavior of a ranked fragmentation.

DOI : 10.1051/ps:2002009
Classification : 60J25, 60G09
Mots clés : fragmentation, self-similar, subordinator, exchangeable partitions, record process 
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Berestycki, Julien. Ranked fragmentations. ESAIM: Probability and Statistics, Tome 6 (2002), pp. 157-175. doi : 10.1051/ps:2002009. http://www.numdam.org/articles/10.1051/ps:2002009/

[1] D.J. Aldous, Exchangeability and related topics, edited by P.L. Hennequin, Lectures on probability theory and statistics, École d'été de Probabilité de Saint-Flour XIII. Springer, Berlin, Lectures Notes in Math. 1117 (1985). | Zbl

[2] D.J. Aldous, Deterministic and stochastic models for coalescence (aggregation and coagulation): A review of the mean-field theory for probabilists. Bernoulli 5 (1999) 3-48. | MR | Zbl

[3] D.J. Aldous and J. Pitman, The standard additive coalescent. Ann. Probab. 26 (1998) 1703-1726. | MR | Zbl

[4] J. Bertoin, Lévy processes. Cambridge University Press, Cambridge (1996). | MR | Zbl

[5] J. Bertoin, Homogeneous fragmentation processes. Probab. Theory Related Fields 121 (2001) 301-318. | MR | Zbl

[6] J. Bertoin, Self-similar fragmentations. Ann. Inst. H. Poincaré (to appear). | Numdam | MR | Zbl

[7] J. Bertoin, The asymptotic behaviour of fragmentation processes, Prépublication du Laboratoire de Probabilités et Modèles Aléatoires, Paris 6 et 7. PMA-651 (2001). | Zbl

[8] N.H. Bingham, C.M. Goldie and J.L. Teugels, Regular variation. Cambridge University Press, Encyclopedia Math. Appl. 27 (1987). | MR | Zbl

[9] E. Bolthausen and A.S. Sznitman, On Ruelle's probability cascades and an abstract cavity method. Commun. Math. Phys. 197 (1998) 247-276. | Zbl

[10] M.D. Brennan and R. Durrett, Splitting intervals. Ann. Probab. 14 (1986) 1024-1036. | MR | Zbl

[11] M.D. Brennan and R. Durrett, Splitting intervals II. Limit laws for lengths. Probab. Theory Related Fields 75 (1987) 109-127. | MR | Zbl

[12] C. Dellacherie and P. Meyer, Probabilités et potentiel, Chapitres V à VIII. Hermann, Paris (1980). | MR | Zbl

[13] S.N. Evans and J. Pitman, Construction of Markovian coalescents. Ann. Inst. H. Poincaré Probab. Statist. 34 (1998) 339-383. | Numdam | MR | Zbl

[14] N. Ikeda and S. Watanabe, Stochastic Differential Equations and Diffusion Processes. North-Holland Mathematical Library (1981). | MR | Zbl

[15] J.F.C. Kingman, The coalescent. Stochastic Process. Appl. 13 (1960) 235-248. | MR | Zbl

[16] M. Perman, Order statistics for jumps of normalised subordinators. Stochastic Process. Appl. 46 (1993) 267-281. | MR | Zbl

[17] J. Pitman, Coalescents with multiple collisions. Ann. Probab. 27 (1999) 1870-1902. | MR | Zbl

[18] K. Sato, Lévy Processes and Infinitly Divisible Distributions. Cambridge University Press, Cambridge, Cambridge Stud. Adv. Math. 68 (1999). | MR | Zbl

[19] J. Schweinsberg, Coalescents with simultaneous multiple collisions. Electr. J. Probab. 5-12 (2000) 1-50. http://www.math.washington.edu/ ejpecp.ejp5contents.html | MR | Zbl

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