Bernoulli

  • Bernoulli
  • Volume 17, Number 1 (2011), 395-423.

Statistical analysis of self-similar conservative fragmentation chains

Marc Hoffmann and Nathalie Krell

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Abstract

We explore statistical inference in self-similar conservative fragmentation chains when only approximate observations of the sizes of the fragments below a given threshold are available. This framework, introduced by Bertoin and Martinez [Adv. Appl. Probab. 37 (2005) 553–570], is motivated by mineral crushing in the mining industry. The underlying object that can be identified from the data is the step distribution of the random walk associated with a randomly tagged fragment that evolves along the genealogical tree representation of the fragmentation process. We compute upper and lower rates of estimation in a parametric framework and show that in the nonparametric case, the difficulty of the estimation is comparable to ill-posed linear inverse problems of order 1 in signal denoising.

Article information

Source
Bernoulli, Volume 17, Number 1 (2011), 395-423.

Dates
First available in Project Euclid: 8 February 2011

Permanent link to this document
https://projecteuclid.org/euclid.bj/1297173847

Digital Object Identifier
doi:10.3150/10-BEJ274

Mathematical Reviews number (MathSciNet)
MR2797996

Zentralblatt MATH identifier
1284.62535

Keywords
fragmentation chains key renewal theorem nonparametric estimation parametric

Citation

Hoffmann, Marc; Krell, Nathalie. Statistical analysis of self-similar conservative fragmentation chains. Bernoulli 17 (2011), no. 1, 395--423. doi:10.3150/10-BEJ274. https://projecteuclid.org/euclid.bj/1297173847


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References

  • [1] Aldous, D.J. (1999). Deterministic and stochastic models for coalescence (aggregation, coagulation): A review of the mean-field theory for probabilists. Bernoulli 5 3–48.
  • [2] Asmussen, S. (2003). Applied Probability and Queues, 2nd edition. New York: Springer.
  • [3] Bertoin, J. (1996). Lévy Processes. Cambridge: Cambridge Univ. Press.
  • [4] Bertoin, J. (2002). Self similar fragmentations. Ann. Inst. H. Poincaré. Probab. Statist. 38 319–340.
  • [5] Bertoin, J. (2006). Random Fragmentation and Coagulation Processes. Cambridge: Cambridge Univ. Press.
  • [6] Bertoin, J. and Martinez, S. (2005). Fragmentation energy. Adv. in Appl. Probab. 37 553–570.
  • [7] Brennan, M.D. and Durett, R. (1987). Splitting intervals. II. Limit laws for lengths. Probab. Theory Related Fields 75 109–127.
  • [8] Filippov, A.F. (1961). On the distribution of the sizes of particles which undergo splitting. Theory Probab. Appl. 6 275–293.
  • [9] Hille, E. and Phillips, R.S. (1957). Functional Analysis and Semi-Groups. Amer. Math. Soc. Colloquium Publications 31. Providence, RI: Amer. Math. Soc.
  • [10] Jagers, P. and Nerman, O. (1984). The growth and composition of branching populations. Adv. in Appl. Probab. 16 221–259.
  • [11] Kolmogorov, A.N. (1941). Über das logaritmisch normale Verteinlungsgesetz der Dimensionen des Teilchen bei Zerstückelung. C. R. (Doklady) Acad. Sci. USSR 31 99–101.
  • [12] Krapivsky, P.L. and Ben-Naim, E. (1994). Scaling and multiscaling in models of fragmentation. Phys. Rev. E 50 3502–3507.
  • [13] Krapivsky, P.L., Ben-Naim, E. and Grosse, I. (2004). Stable distributions in stochastic fragmentation. J. Phys. A 37 2863–2880.
  • [14] Ney, P. (1981). A refinement of the coupling method in renewal theory. Stochastic Process. Appl. 11 11–26.
  • [15] Sgibnev, M.S. (2002). Stone’s decomposition of the renewal measure via Banach-algebraic techniques. Proc. Amer. Math. Soc. 130 2425–2430.
  • [16] Tsybakov, A.B. (2009). Introduction to Non-Parametric Estimation. New York: Springer.