• Bernoulli
  • Volume 14, Number 4 (2008), 988-1002.

Gibbs fragmentation trees

Peter McCullagh, Jim Pitman, and Matthias Winkel

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We study fragmentation trees of Gibbs type. In the binary case, we identify the most general Gibbs-type fragmentation tree with Aldous’ beta-splitting model, which has an extended parameter range β>−2 with respect to the beta(β+1, β+1) probability distributions on which it is based. In the multifurcating case, we show that Gibbs fragmentation trees are associated with the two-parameter Poisson–Dirichlet models for exchangeable random partitions of ℕ, with an extended parameter range 0≤α≤1, θ≥−2α and α<0, θ=−, m∈ℕ.

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Bernoulli Volume 14, Number 4 (2008), 988-1002.

First available in Project Euclid: 6 November 2008

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Aldous’ beta-splitting model Gibbs distribution Markov branching model Poisson–Dirichlet distribution


McCullagh, Peter; Pitman, Jim; Winkel, Matthias. Gibbs fragmentation trees. Bernoulli 14 (2008), no. 4, 988--1002. doi:10.3150/08-BEJ134.

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  • [1] Aldous, D. (1991). The continuum random tree. I., Ann. Probab. 19 1–28.
  • [2] Aldous, D. (1996). Probability distributions on cladograms. In, Random Discrete Structures (Minneapolis, MN, 1993). IMA Vol. Math. Appl. 76 1–18. New York: Springer.
  • [3] Berestycki, N. and Pitman, J. (2007). Gibbs distributions for random partitions generated by a fragmentation process., J. Stat. Phys. 127 381–418.
  • [4] Bertoin, J. (2001). Homogeneous fragmentation processes., Probab. Theory Related Fields 121 301–318.
  • [5] Duquesne, T. and Le Gall, J.-F. (2002). Random trees, Lévy processes and spatial branching processes., Astérisque 281 vi+147.
  • [6] Ford, D.J. (2005). Probabilities on cladograms: Introduction to the alpha model. Preprint., arXiv:math.PR/0511246.
  • [7] Gnedin, A. and Pitman, J. (2005). Exchangeable Gibbs partitions and Stirling triangles., Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 325 (Teor. Predst. Din. Sist. Komb. i Algoritm. Metody 12) 83–102, 244–245.
  • [8] Gnedin, A. and Pitman, J. (2006). Moments of convex distribution functions and completely alternating sequences. Preprint., arXiv:math.PR/0602091.
  • [9] Haas, B., Miermont, G., Pitman, J. and Winkel, M. (2006). Continuum tree asymptotics of discrete fragmentations and applications to phylogenetic models. Preprint. arXiv:math.PR/0604350., Ann. Probab. To appear.
  • [10] Haas, B., Pitman, J. and Winkel, M. (2007). Spinal partitions and invariance under re-rooting of continuum random trees. Preprint. arXiv:0705.3602., Ann. Probab. To appear.
  • [11] Kerov, S. (2005). Coherent random allocations, and the Ewens–Pitman formula., Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 325 (Teor. Predst. Din. Sist. Komb. i Algoritm. Metody 12) 127–145, 246.
  • [12] McCullagh, P., Pitman, J. and Winkel, M. (2007). Gibbs fragmentation trees. Preprint., arXiv:0704.0945.
  • [13] Miermont, G. (2003). Self-similar fragmentations derived from the stable tree. I. Splitting at heights., Probab. Theory Related Fields 127 423–454.
  • [14] Pitman, J. (2003). Poisson–Kingman partitions. In, Statistics and Science: A Festschrift for Terry Speed. IMS Lecture Notes Monogr. Ser. 40 1–34. Beachwood, OH: Inst. Math. Statist.
  • [15] Pitman, J. (2006)., Combinatorial Stochastic Processes. Lecture Notes in Math. 1875. Lectures from the 32nd Summer School on Probability Theory held in Saint-Flour, July 7–24, 2002. Berlin: Springer.
  • [16] Schroeder, E. (1870). Vier combinatorische Probleme., Z. f. Math. Phys. 15 361–376.
  • [17] Semple, C. and Steel, M. (2003)., Phylogenetics. Oxford Lecture Series in Mathematics and Its Applications 24. Oxford Univ. Press.
  • [18] Stanley, R.P. (1999)., Enumerative Combinatorics. 2. Cambridge Studies in Advanced Mathematics 62. Cambridge Univ. Press.