## Electronic Journal of Probability

### Regenerative tree growth: structural results and convergence

#### Abstract

We introduce regenerative tree growth processes as consistent families of random trees with n labelled leaves, n>=1, with a regenerative property at branch points. This framework includes growth processes for exchangeably labelled Markov branching trees, as well as non-exchangeable models such as the alpha-theta model, the alpha-gamma model and all restricted exchangeable models previously studied. Our main structural result is a representation of the growth rule by a sigma-finite dislocation measure kappa on the set of partitions of the natural numbers extending Bertoin's notion of exchangeable dislocation measures from the setting of homogeneous fragmentations. We use this representation to establish necessary and sufficient conditions on the growth rule under which we can apply results by Haas and Miermont for unlabelled and not necessarily consistent trees to establish self-similar random trees and residual mass processes as scaling limits. While previous studies exploited some form of exchangeability, our scaling limit results here only require a regularity condition on the convergence of asymptotic frequencies under kappa, in addition to a regular variation condition.

#### Article information

Source
Electron. J. Probab., Volume 19 (2014), paper no. 70, 27 pp.

Dates
Accepted: 15 August 2014
First available in Project Euclid: 4 June 2016

https://projecteuclid.org/euclid.ejp/1465065712

Digital Object Identifier
doi:10.1214/EJP.v19-3040

Mathematical Reviews number (MathSciNet)
MR3256870

Zentralblatt MATH identifier
1304.60096

Rights

#### Citation

Pitman, Jim; Rizzolo, Douglas; Winkel, Matthias. Regenerative tree growth: structural results and convergence. Electron. J. Probab. 19 (2014), paper no. 70, 27 pp. doi:10.1214/EJP.v19-3040. https://projecteuclid.org/euclid.ejp/1465065712

#### References

• Aldous, David. The continuum random tree. I. Ann. Probab. 19 (1991), no. 1, 1–28.
• Aldous, David. The continuum random tree. III. Ann. Probab. 21 (1993), no. 1, 248–289.
• Aldous, David. Probability distributions on cladograms. Random discrete structures (Minneapolis, MN, 1993), 1–18, IMA Vol. Math. Appl., 76, Springer, New York, 1996.
• 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.
• Chen, Bo; Ford, Daniel; Winkel, Matthias. A new family of Markov branching trees: the alpha-gamma model. Electron. J. Probab. 14 (2009), no. 15, 400–430.
• Chen, Bo; Winkel, Matthias. Restricted exchangeable partitions and embedding of associated hierarchies in continuum random trees. Ann. Inst. Henri Poincaré Probab. Stat. 49 (2013), no. 3, 839–872.
• Duquesne, Thomas; Le Gall, Jean-François. (2002). Random trees, Lévy processes and spatial branching processes, Astérisque, no. 281, vi+147.
• Evans, Steven N.; Pitman, Jim; Winter, Anita. Rayleigh processes, real trees, and root growth with re-grafting. Probab. Theory Related Fields 134 (2006), no. 1, 81–126.
• Evans, Steven N.; Winter, Anita. Subtree prune and regraft: a reversible real tree-valued Markov process. Ann. Probab. 34 (2006), no. 3, 918–961.
• Ford, Daniel J. Probabilities on cladograms: Introduction to the alpha model. Thesis (Ph.D.)€- Stanford University. ProQuest LLC, Ann Arbor, MI, 2006. 241 pp. ISBN: 978-0542-70672-1.
• Gnedin, Alexander; Pitman, Jim. Regenerative composition structures. Ann. Probab. 33 (2005), no. 2, 445–479.
• Greven, Andreas; Pfaffelhuber, Peter; Winter, Anita. Convergence in distribution of random metric measure spaces ($\Lambda$-coalescent measure trees). Probab. Theory Related Fields 145 (2009), no. 1-2, 285–322.
• Gromov, Mischa. (1999). Metric structures for Riemannian and non-Riemannian spaces, Progress in Mathematics, vol. 152, BirkhÃ¤user Boston Inc., Boston, MA, Based on the 1981 French original, with appendices by M. Katz, P. Pansu and S. Semmes, Translated from the French by Sean Michael Bates.
• Haas, Bénédicte; Miermont, Grégory. The genealogy of self-similar fragmentations with negative index as a continuum random tree. Electron. J. Probab. 9 (2004), no. 4, 57–97 (electronic).
• Haas, Bénédicte; Miermont, Gréory. Scaling limits of Markov branching trees with applications to Galton-Watson and random unordered trees. Ann. Probab. 40 (2012), no. 6, 2589–2666.
• 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.
• Haas, Bénédicte; Pitman, Jim; Winkel, Matthias. Spinal partitions and invariance under re-rooting of continuum random trees. Ann. Probab. 37 (2009), no. 4, 1381–1411.
• Haas, Bénédicte; Miermont, Grégory. Self-similar scaling limits of non-increasing Markov chains. Bernoulli 17 (2011), no. 4, 1217–1247.
• Haulk, Chris; Pitman, Jim. (2011). phA representation of exchangeable hierarchies by sampling from real trees, arXiv:1101.5619 [math.PR].
• Lamperti, John. Semi-stable Markov processes. I. Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 22 (1972), 205–225.
• Marchal, Philippe. A note on the fragmentation of a stable tree. Fifth Colloquium on Mathematics and Computer Science, 489–499, Discrete Math. Theor. Comput. Sci. Proc., AI, Assoc. Discrete Math. Theor. Comput. Sci., Nancy, 2008.
• McCullagh, Peter; Pitman, Jim; Winkel, Matthias. Gibbs fragmentation trees. Bernoulli 14 (2008), no. 4, 988–1002.
• Miermont, Grégory. Self-similar fragmentations derived from the stable tree. I. Splitting at heights. Probab. Theory Related Fields 127 (2003), no. 3, 423–454.
• Miermont, Grégory. Tessellations of random maps of arbitrary genus. Ann. Sci. Éc. Norm. Supér. (4) 42 (2009), no. 5, 725–781.
• Pal, Soumik. (2011). On the Aldous diffusion on Continuum Trees. I, arXiv:1104.4186 [math.PR].
• Pitman, Jim. Combinatorial stochastic processes. Lectures from the 32nd Summer School on Probability Theory held in Saint-Flour, July 7 - 24, 2002. With a foreword by Jean Picard. Lecture Notes in Mathematics, 1875. Springer-Verlag, Berlin, 2006. x+256 pp. ISBN: 978-3-540-30990-1; 3-540-30990-X.
• Pitman, Jim. Exchangeable and partially exchangeable random partitions. Probab. Theory Related Fields 102 (1995), no. 2, 145–158.
• Rizzolo, Douglas Paul. Scaling limits of random trees. Thesis (Ph.D.) - University of California, Berkeley. ProQuest LLC, Ann Arbor, MI, 2012. 61 pp. ISBN: 978-1267-61328-8.
• Pitman, Jim; Winkel, Matthias. Regenerative tree growth: binary self-similar continuum random trees and Poisson-Dirichlet compositions. Ann. Probab. 37 (2009), no. 5, 1999–2041.
• Pitman, Jim; Winkel, Matthias. (2013). Regenerative tree growth: Markovian embedding of fragmenters, bifurcators and bead splitting processes, to appear in Ann. Prob., arXiv:1304.0802 [math.PR].