Consistent Markov branching trees with discrete edge lengths

Harry Crane (Rutgers University)


We study consistent collections of random fragmentation trees with random integer-valued edge lengths. We prove several equivalent necessary and sufficient conditions under which Geometrically distributed edge lengths can be consistently assigned to a Markov branching tree. Among these conditions is a characterization by a unique probability measure, which plays a role similar to the dislocation measure for homogeneous fragmentation processes. We discuss this and other connections to previous work on Markov branching trees and homogeneous fragmentation processes.

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Pages: 1-14

Publication Date: August 31, 2013

DOI: 10.1214/ECP.v18-2872


  • Aldous, David. Probability distributions on cladograms. Random discrete structures (Minneapolis, MN, 1993), 1--18, IMA Vol. Math. Appl., 76, Springer, New York, 1996. MR1395604
  • Bertoin, Jean. Homogeneous fragmentation processes. Probab. Theory Related Fields 121 (2001), no. 3, 301--318. MR1867425
  • Bertoin, Jean. Random fragmentation and coagulation processes. Cambridge Studies in Advanced Mathematics, 102. Cambridge University Press, Cambridge, 2006. viii+280 pp. ISBN: 978-0-521-86728-3; 0-521-86728-2 MR2253162
  • Haas, Bénédicte; Miermont, Grégory; Pitman, Jim; Winkel, Matthias. Continuum tree asymptotics of discrete fragmentations and applications to phylogenetic models. Ann. Probab. 36 (2008), no. 5, 1790--1837. MR2440924
  • Kingman, J. F. C. The representation of partition structures. J. London Math. Soc. (2) 18 (1978), no. 2, 374--380. MR0509954
  • McCullagh, Peter; Pitman, Jim; Winkel, Matthias. Gibbs fragmentation trees. Bernoulli 14 (2008), no. 4, 988--1002. MR2543583
  • Tavaré, Simon. Ancestral inference in population genetics. Lectures on probability theory and statistics, 1--188, Lecture Notes in Math., 1837, Springer, Berlin, 2004. MR2071630

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