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

Christina Goldschmidt; Bénédicte Haas

Annales de l'I.H.P. Probabilités et statistiques (2010)

  • Volume: 46, Issue: 2, page 338-368
  • ISSN: 0246-0203

Abstract

top
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.

How to cite

top

Goldschmidt, Christina, and Haas, Bénédicte. "Behavior near the extinction time in self-similar fragmentations I : the stable case." Annales de l'I.H.P. Probabilités et statistiques 46.2 (2010): 338-368. <http://eudml.org/doc/239331>.

@article{Goldschmidt2010,
abstract = {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.},
author = {Goldschmidt, Christina, Haas, Bénédicte},
journal = {Annales de l'I.H.P. Probabilités et statistiques},
keywords = {stable Lévy processes; height processes; self-similar fragmentations; extinction time; scaling limits},
language = {eng},
number = {2},
pages = {338-368},
publisher = {Gauthier-Villars},
title = {Behavior near the extinction time in self-similar fragmentations I : the stable case},
url = {http://eudml.org/doc/239331},
volume = {46},
year = {2010},
}

TY - JOUR
AU - Goldschmidt, Christina
AU - Haas, Bénédicte
TI - Behavior near the extinction time in self-similar fragmentations I : the stable case
JO - Annales de l'I.H.P. Probabilités et statistiques
PY - 2010
PB - Gauthier-Villars
VL - 46
IS - 2
SP - 338
EP - 368
AB - 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.
LA - eng
KW - stable Lévy processes; height processes; self-similar fragmentations; extinction time; scaling limits
UR - http://eudml.org/doc/239331
ER -

References

top
  1. [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. Zbl1162.60326MR2508567
  2. [2] A.-L. Basdevant. Fragmentation of ordered partitions and intervals. Electron. J. Probab. 11 (2006) 394–417 (electronic). Zbl1109.60058MR2223041
  3. [3] J. Bertoin. Lévy Processes. Cambridge Tracts in Mathematics 121. Cambridge Univ. Press, Cambridge, 1996. Zbl0861.60003MR1406564
  4. [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. Zbl0955.60046MR1746300
  5. [5] J. Bertoin. Self-similar fragmentations. Ann. Inst. H. Poincaré Probab. Statist. 38 (2002) 319–340. Zbl1002.60072MR1899456
  6. [6] J. Bertoin. The asymptotic behavior of fragmentation processes. J. Eur. Math. Soc. (JEMS) 5 (2003) 395–416. Zbl1042.60042MR2017852
  7. [7] J. Bertoin. Random Fragmentation and Coagulation Processes. Cambridge Studies in Advanced Mathematics 102. Cambridge Univ. Press, Cambridge, 2006. Zbl1107.60002MR2253162
  8. [8] J. Bertoin and A. Rouault. Discretization methods for homogeneous fragmentations. J. London Math. Soc. (2) 72 (2005) 91–109. Zbl1077.60053MR2145730
  9. [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). Zbl1040.11061MR1848256
  10. [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. Zbl0905.60056MR1648657
  11. [11] T. Duquesne and J.-F. Le Gall. Random trees, Lévy processes and spatial branching processes. Astérisque 281 (2002) 1–147. Zbl1037.60074MR1954248
  12. [12] C. Goldschmidt and B. Haas. Behavior near the extinction time in self-similar fragmentations II: Finite dislocation measures. In preparation, 2010. Zbl06571515
  13. [13] B. Haas. Loss of mass in deterministic and random fragmentations. Stochastic Process. Appl. 106 (2003) 245–277. Zbl1075.60553MR1989629
  14. [14] B. Haas. Regularity of formation of dust in self-similar fragmentations. Ann. Inst. H. Poincaré Probab. Statist. 40 (2004) 411–438. Zbl1041.60058MR2070333
  15. [15] B. Haas. Fragmentation processes with an initial mass converging to infinity. J. Theoret. Probab. 20 (2007) 721–758. Zbl1135.60048MR2359053
  16. [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). Zbl1064.60076MR2041829
  17. [17] J. Jacod and A. N. Shiryaev. Limit Theorems for Stochastic Processes, 2nd edition. Grundlehren der Mathematischen Wissenschaften 288. Springer, Berlin, 2003. Zbl1018.60002MR1943877
  18. [18] D. P. Kennedy. The distribution of the maximum Brownian excursion. J. Appl. Probab. 13 (1976) 371–376. Zbl0338.60048MR402955
  19. [19] J.-F. Le Gall and Y. Le Jan. Branching processes in Lévy processes: The exploration process. Ann. Probab. 26 (1998) 213–252. Zbl0948.60071MR1617047
  20. [20] G. Miermont. Self-similar fragmentations derived from the stable tree. I. Splitting at heights. Probab. Theory Related Fields 127 (2003) 423–454. Zbl1042.60043MR2018924
  21. [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. Zbl1103.60004MR2245368
  22. [22] L. C. G. Rogers and D. Williams. Diffusions, Markov Processes and Martingales, Vol. 2: Itô Calculus. Cambridge Univ. Press, Cambridge, 2000. Zbl0977.60005MR1780932
  23. [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. Zbl1208.60036MR2572168

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.