Ranked fragmentations

Julien Berestycki

ESAIM: Probability and Statistics (2002)

  • Volume: 6, page 157-175
  • ISSN: 1292-8100

Abstract

top
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.

How to cite

top

Berestycki, Julien. "Ranked fragmentations." ESAIM: Probability and Statistics 6 (2002): 157-175. <http://eudml.org/doc/245516>.

@article{Berestycki2002,
abstract = {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.},
author = {Berestycki, Julien},
journal = {ESAIM: Probability and Statistics},
keywords = {fragmentation; self-similar; subordinator; exchangeable partitions; record process},
language = {eng},
pages = {157-175},
publisher = {EDP-Sciences},
title = {Ranked fragmentations},
url = {http://eudml.org/doc/245516},
volume = {6},
year = {2002},
}

TY - JOUR
AU - Berestycki, Julien
TI - Ranked fragmentations
JO - ESAIM: Probability and Statistics
PY - 2002
PB - EDP-Sciences
VL - 6
SP - 157
EP - 175
AB - 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.
LA - eng
KW - fragmentation; self-similar; subordinator; exchangeable partitions; record process
UR - http://eudml.org/doc/245516
ER -

References

top
  1. [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). Zbl0562.60042
  2. [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. Zbl0930.60096MR1673235
  3. [3] D.J. Aldous and J. Pitman, The standard additive coalescent. Ann. Probab. 26 (1998) 1703-1726. Zbl0936.60064MR1675063
  4. [4] J. Bertoin, Lévy processes. Cambridge University Press, Cambridge (1996). Zbl0861.60003MR1406564
  5. [5] J. Bertoin, Homogeneous fragmentation processes. Probab. Theory Related Fields 121 (2001) 301-318. Zbl0992.60076MR1867425
  6. [6] J. Bertoin, Self-similar fragmentations. Ann. Inst. H. Poincaré (to appear). Zbl1002.60072MR1899456
  7. [7] J. Bertoin, The asymptotic behaviour of fragmentation processes, Prépublication du Laboratoire de Probabilités et Modèles Aléatoires, Paris 6 et 7. PMA-651 (2001). Zbl1042.60042
  8. [8] N.H. Bingham, C.M. Goldie and J.L. Teugels, Regular variation. Cambridge University Press, Encyclopedia Math. Appl. 27 (1987). Zbl0617.26001MR898871
  9. [9] E. Bolthausen and A.S. Sznitman, On Ruelle’s probability cascades and an abstract cavity method. Commun. Math. Phys. 197 (1998) 247-276. Zbl0927.60071
  10. [10] M.D. Brennan and R. Durrett, Splitting intervals. Ann. Probab. 14 (1986) 1024-1036. Zbl0601.60028MR841602
  11. [11] M.D. Brennan and R. Durrett, Splitting intervals II. Limit laws for lengths. Probab. Theory Related Fields 75 (1987) 109-127. Zbl0607.60010MR879556
  12. [12] C. Dellacherie and P. Meyer, Probabilités et potentiel, Chapitres V à VIII. Hermann, Paris (1980). Zbl0464.60001MR566768
  13. [13] S.N. Evans and J. Pitman, Construction of Markovian coalescents. Ann. Inst. H. Poincaré Probab. Statist. 34 (1998) 339-383. Zbl0906.60058MR1625867
  14. [14] N. Ikeda and S. Watanabe, Stochastic Differential Equations and Diffusion Processes. North-Holland Mathematical Library (1981). Zbl0495.60005MR1011252
  15. [15] J.F.C. Kingman, The coalescent. Stochastic Process. Appl. 13 (1960) 235-248. Zbl0491.60076MR671034
  16. [16] M. Perman, Order statistics for jumps of normalised subordinators. Stochastic Process. Appl. 46 (1993) 267-281. Zbl0777.60070MR1226412
  17. [17] J. Pitman, Coalescents with multiple collisions. Ann. Probab. 27 (1999) 1870-1902. Zbl0963.60079MR1742892
  18. [18] K. Sato, Lévy Processes and Infinitly Divisible Distributions. Cambridge University Press, Cambridge, Cambridge Stud. Adv. Math. 68 (1999). Zbl0973.60001MR1739520
  19. [19] J. Schweinsberg, Coalescents with simultaneous multiple collisions. Electr. J. Probab. 5-12 (2000) 1-50. http://www.math.washington.edu/ ejpecp.ejp5contents.html Zbl0959.60065MR1781024

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.