The Annals of Probability

Behavior near the extinction time in self-similar fragmentations II: Finite dislocation measures

Christina Goldschmidt and Bénédicte Haas

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text


We study a Markovian model for the random fragmentation of an object. At each time, the state consists of a collection of blocks. Each block waits an exponential amount of time with parameter given by its size to some power $\alpha$, independently of the other blocks. Every block then splits randomly into sub-blocks whose relative sizes are distributed according to the so-called dislocation measure. We focus here on the case where $\alpha<0$. In this case, small blocks split intensively, and so the whole state is reduced to “dust” in a finite time, almost surely (we call this the extinction time). In this paper, we investigate how the fragmentation process behaves as it approaches its extinction time. In particular, we prove a scaling limit for the block sizes which, as a direct consequence, gives us an expression for an invariant measure for the fragmentation process. In an earlier paper [Ann. Inst. Henri Poincaré Probab. Stat. 46 (2010) 338–368], we considered the same problem for another family of fragmentation processes, the so-called stable fragmentations. The results here are similar, but we emphasize that the methods used to prove them are different. Our approach in the present paper is based on Markov renewal theory and involves a somewhat unusual “spine” decomposition for the fragmentation, which may be of independent interest.

Article information

Ann. Probab. Volume 44, Number 1 (2016), 739-805.

Received: September 2013
Revised: September 2014
First available in Project Euclid: 2 February 2016

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60J25: Continuous-time Markov processes on general state spaces 60G18: Self-similar processes

Self-similar fragmentations extinction time scaling limits invariant measure Markov renewal theory spine decomposition


Goldschmidt, Christina; Haas, Bénédicte. Behavior near the extinction time in self-similar fragmentations II: Finite dislocation measures. Ann. Probab. 44 (2016), no. 1, 739--805. doi:10.1214/14-AOP988.

Export citation


  • [1] Aldous, D. and Shields, P. (1988). A diffusion limit for a class of randomly-growing binary trees. Probab. Theory Related Fields 79 509–542.
  • [2] Aldous, D. J. and Bandyopadhyay, A. (2005). A survey of max-type recursive distributional equations. Ann. Appl. Probab. 15 1047–1110.
  • [3] Alsmeyer, G. (1994). On the Markov renewal theorem. Stochastic Process. Appl. 50 37–56.
  • [4] Alsmeyer, G. (1997). The Markov renewal theorem and related results. Markov Process. Related Fields 3 103–127.
  • [5] Athreya, K. B. (1985). Discounted branching random walks. Adv. in Appl. Probab. 17 53–66.
  • [6] Athreya, K. B., McDonald, D. and Ney, P. (1978). Limit theorems for semi-Markov processes and renewal theory for Markov chains. Ann. Probab. 6 788–797.
  • [7] Barlow, M. T., Pemantle, R. and Perkins, E. A. (1997). Diffusion-limited aggregation on a tree. Probab. Theory Related Fields 107 1–60.
  • [8] Bertoin, J. (2001). Homogeneous fragmentation processes. Probab. Theory Related Fields 121 301–318.
  • [9] Bertoin, J. (2002). Self-similar fragmentations. Ann. Inst. Henri Poincaré Probab. Stat. 38 319–340.
  • [10] Bertoin, J. (2003). The asymptotic behavior of fragmentation processes. J. Eur. Math. Soc. (JEMS) 5 395–416.
  • [11] Bertoin, J. (2006). Random Fragmentation and Coagulation Processes. Cambridge Studies in Advanced Mathematics 102. Cambridge Univ. Press, Cambridge.
  • [12] Dean, D. S. and Majumdar, S. N. (2006). Phase transition in a generalized Eden growth model on a tree. J. Stat. Phys. 124 1351–1376.
  • [13] Devroye, L. (1986). A note on the height of binary search trees. J. Assoc. Comput. Mach. 33 489–498.
  • [14] Duquesne, T. and Le Gall, J.-F. (2002). Random trees, Lévy processes and spatial branching processes. Astérisque 281 vi+147.
  • [15] Duquesne, T. and Le Gall, J.-F. (2005). Probabilistic and fractal aspects of Lévy trees. Probab. Theory Related Fields 131 553–603.
  • [16] Ethier, S. N. and Kurtz, T. G. (1986). Markov Processes: Characterization and Convergence. Wiley, New York.
  • [17] Filippov, A. F. (1961). On the distribution of the sizes of particles which undergo splitting. Theory Probab. Appl. 6 275–294.
  • [18] Goldschmidt, C. and Haas, B. (2010). Behavior near the extinction time in self-similar fragmentations. I. The stable case. Ann. Inst. Henri Poincaré Probab. Stat. 46 338–368.
  • [19] Haas, B. (2003). Loss of mass in deterministic and random fragmentations. Stochastic Process. Appl. 106 245–277.
  • [20] Haas, B. (2004). Regularity of formation of dust in self-similar fragmentations. Ann. Inst. Henri Poincaré Probab. Stat. 40 411–438.
  • [21] Jacod, J. (1971). Théorème de renouvellement et classification pour les chaînes semi-markoviennes. Ann. Inst. H. Poincaré Sect. B (N.S.) 7 83–129.
  • [22] Kesten, H. (1974). Renewal theory for functionals of a Markov chain with general state space. Ann. Probab. 2 355–386.
  • [23] Meyn, S. P. and Tweedie, R. L. (1993). Markov Chains and Stochastic Stability. Springer London, London.
  • [24] Miermont, G. (2003). Self-similar fragmentations derived from the stable tree. I. Splitting at heights. Probab. Theory Related Fields 127 423–454.
  • [25] Orey, S. (1961). Change of time scale for Markov processes. Trans. Amer. Math. Soc. 99 384–397.
  • [26] Pittel, B. (1984). On growing random binary trees. J. Math. Anal. Appl. 103 461–480.
  • [27] Roberts, G. O. and Rosenthal, J. S. (2004). General state space Markov chains and MCMC algorithms. Probab. Surv. 1 20–71.
  • [28] Shurenkov, V. (1985). On the theory of Markov renewal. Theory Probab. Appl. 29 247–265.