Electronic Journal of Probability

Totally ordered measured trees and splitting trees with infinite variation

Amaury Lambert and Gerónimo Uribe Bravo

Full-text: Open access

Abstract

Combinatorial trees can be used to represent genealogies of asexual individuals. These individuals can be endowed with birth and death times, to obtain a so-called ‘chronological tree’. In this work, we are interested in the continuum analogue of chronological trees in the setting of real trees. This leads us to consider totally ordered and measured trees, abbreviated as TOM trees. First, we define an adequate space of TOM trees and prove that under some mild conditions, every compact TOM tree can be represented in a unique way by a so-called contour function, which is right-continuous, admits limits from the left and has non-negative jumps. The appropriate notion of contour function is also studied in the case of locally compact TOM trees. Then we study the splitting property of (measures on) TOM trees which extends the notion of ‘splitting tree’ studied in [Lam10], where during her lifetime, each individual gives birth at constant rate to independent and identically distributed copies of herself. We prove that the contour function of a TOM tree satisfying the splitting property is associated to a spectrally positive Lévy process that is not a subordinator, both in the critical and subcritical cases of compact trees as well as in the supercritical case of locally compact trees. The genealogical trees associated to splitting trees are the celebrated Lévy trees in the subcritical case and they will be analyzed in the supercritical case in forthcoming work.

Article information

Source
Electron. J. Probab., Volume 23 (2018), paper no. 120, 41 pp.

Dates
Received: 9 June 2017
Accepted: 24 November 2018
First available in Project Euclid: 15 December 2018

Permanent link to this document
https://projecteuclid.org/euclid.ejp/1544843300

Digital Object Identifier
doi:10.1214/18-EJP251

Mathematical Reviews number (MathSciNet)
MR3896857

Zentralblatt MATH identifier
07021676

Subjects
Primary: 60G51: Processes with independent increments; Lévy processes 60J80: Branching processes (Galton-Watson, birth-and-death, etc.) 05C05: Trees 92D25: Population dynamics (general)

Keywords
chronological trees Lévy processes splitting property

Rights
Creative Commons Attribution 4.0 International License.

Citation

Lambert, Amaury; Uribe Bravo, Gerónimo. Totally ordered measured trees and splitting trees with infinite variation. Electron. J. Probab. 23 (2018), paper no. 120, 41 pp. doi:10.1214/18-EJP251. https://projecteuclid.org/euclid.ejp/1544843300


Export citation

References

  • [AD12] Romain Abraham and Jean-François Delmas, A continuum-tree-valued Markov process, Ann. Probab. 40 (2012), no. 3, 1167–1211.
  • [ADH14] Romain Abraham, Jean-François Delmas, and Patrick Hoscheit, Exit times for an increasing Lévy tree-valued process, Probab. Theory Related Fields 159 (2014), no. 1-2, 357–403.
  • [Ald91a] David Aldous, The continuum random tree. I, Ann. Probab. 19 (1991), no. 1, 1–28.
  • [Ald91b] David Aldous, The continuum random tree. II. An overview, Stochastic analysis (Durham, 1990), London Math. Soc. Lecture Note Ser., vol. 167, Cambridge Univ. Press, Cambridge, 1991, pp. 23–70.
  • [Ald93] David Aldous, The continuum random tree. III, Ann. Probab. 21 (1993), no. 1, 248–289.
  • [BBI01] Dmitri Burago, Yuri Burago, and Sergei Ivanov, A course in metric geometry, Graduate Studies in Mathematics, vol. 33, American Mathematical Society, Providence, RI, 2001.
  • [Ber91] Jean Bertoin, Sur la décomposition de la trajectoire d’un processus de Lévy spectralement positif en son infimum, Ann. Inst. H. Poincaré Probab. Statist. 27 (1991), no. 4, 537–547.
  • [Ber96] Jean Bertoin, Lévy processes, Cambridge Tracts in Mathematics, vol. 121, Cambridge University Press, Cambridge, 1996.
  • [Ber99] Jean Bertoin, Subordinators: examples and applications, Lecture Notes in Math., vol. 1717, pp. 1–91, Springer, Berlin, 1999.
  • [BFM08] Jean Bertoin, Joaquin Fontbona, and Servet Martínez, On prolific individuals in a supercritical continuous-state branching process, J. Appl. Probab. 45 (2008), no. 3, 714–726.
  • [BKMS11] J. Berestycki, A. E. Kyprianou, and A. Murillo-Salas, The prolific backbone for supercritical superprocesses, Stochastic Process. Appl. 121 (2011), no. 6, 1315–1331.
  • [BO18] Gabriel Hernán Berzunza Ojeda, On scaling limits of multitype Galton-Watson trees with possibly infinite variance, ALEA Lat. Am. J. Probab. Math. Stat. 15 (2018), no. 1, 21–48.
  • [Cha13] L. Chaumont, On the law of the supremum of Lévy processes, Ann. Probab. 41 (2013), no. 3A, 1191–1217.
  • [Del08] J.-F. Delmas, Height process for super-critical continuous state branching process, Markov Process. Related Fields 14 (2008), no. 2, 309–326.
  • [DLG02] Thomas Duquesne and Jean-François Le Gall, Random trees, Lévy processes and spatial branching processes, Astérisque (2002), no. 281, vi+147.
  • [DLG05] Thomas Duquesne and Jean-François Le Gall, Probabilistic and fractal aspects of Lévy trees, Probab. Theory Related Fields 131 (2005), no. 4, 553–603.
  • [Don07] Ronald A. Doney, Fluctuation theory for Lévy processes, Lecture Notes in Mathematics, vol. 1897, Springer, Berlin, 2007.
  • [DT96] A. W. M. Dress and W. F. Terhalle, The real tree, Adv. Math. 120 (1996), no. 2, 283–301.
  • [Duq08] Thomas Duquesne, The coding of compact real trees by real valued functions, http://arxiv.org/abs/math/0604106, 2008.
  • [DW07] Thomas Duquesne and Matthias Winkel, Growth of Lévy trees, Probab. Theory Related Fields 139 (2007), no. 3-4, 313–371.
  • [Dyn65] E. B. Dynkin, Markov processes. Vols. I, II, Die Grundlehren der Mathematischen Wissenschaften, vol. 122, Academic Press Inc.; Springer-Verlag, 1965.
  • [EPW06] Steven N. Evans, Jim Pitman, and Anita Winter, Rayleigh processes, real trees, and root growth with re-grafting, Probab. Theory Related Fields 134 (2006), no. 1, 81–126.
  • [GK97] J. Geiger and G. Kersting, Depth-first search of random trees, and Poisson point processes, Classical and modern branching processes (Minneapolis, MN, 1994), IMA Vol. Math. Appl., vol. 84, Springer, New York, 1997, pp. 111–126.
  • [Gro07] Misha Gromov, Metric structures for Riemannian and non-Riemannian spaces, Birkhäuser, 2007.
  • [Itô72] Kiyosi Itô, Poisson point processes attached to Markov processes, Proceedings of the Sixth Berkeley Symposium on Mathematical Statistics and Probability (Univ. California, Berkeley, Calif., 1970/1971), Vol. III: Probability theory (Berkeley, Calif.), Univ. California Press, 1972, pp. 225–239.
  • [Kal02] Olav Kallenberg, Foundations of modern probability, 2nd ed., Springer-Verlag, 2002.
  • [Lam08] Amaury Lambert, Population dynamics and random genealogies, Stoch. Models 24 (2008), no. suppl. 1, 45–163.
  • [Lam10] Amaury Lambert, The contour of splitting trees is a Lévy process, Ann. Probab. 38 (2010), no. 1, 348–395.
  • [Lam17] Amaury Lambert, Probabilistic models for the (sub)tree(s) of life, Braz. J. Probab. Stat. 31 (2017), no. 3, 415–475.
  • [LG99] Jean-François Le Gall, Spatial branching processes, random snakes and partial differential equations, Lectures in Mathematics ETH Zürich, Birkhäuser Verlag, Basel, 1999.
  • [LG06] Jean-François Le Gall, Random real trees, Ann. Fac. Sci. Toulouse Math. (6) 15 (2006), no. 1, 35–62.
  • [LGLJ98] Jean-Francois Le Gall and Yves Le Jan, Branching processes in Lévy processes: the exploration process, Ann. Probab. 26 (1998), no. 1, 213–252.
  • [LU18] A. Lambert and G. Uribe Bravo, Totally Ordered Measured Trees and Splitting Trees with Infinite Variation II: Prolific Skeleton Decomposition, ArXiv e-prints (2018), 1803.05421.
  • [Par05] K. R. Parthasarathy, Probability measures on metric spaces, AMS Chelsea Publishing, Providence, RI, 2005, Reprint of the 1967 original.
  • [PR69] E. A. Pecherskii and B. A. Rogozin, On joint distributions of random variables associated with fluctuations of a process with independent increments, Theory of Probability and its Applications 14 (1969), no. 3, 410–14.
  • [PUB12] Jim Pitman and Gerónimo Uribe Bravo, The convex minorant of a Lévy process, Ann. Probab. 40 (2012), no. 4, 1636–1674.
  • [RY99] Daniel Revuz and Marc Yor, Continuous martingales and Brownian motion, 3rd ed., Grundlehren der Mathematischen Wissenschaften, vol. 293, Springer-Verlag, Berlin, 1999.
  • [Sat99] Ken-iti Sato, Lévy processes and infinitely divisible distributions, Cambridge Studies in Advanced Mathematics, vol. 68, Cambridge University Press, Cambridge, 1999.
  • [Sha88] Michael Sharpe, General theory of Markov processes, Pure and Applied Mathematics, vol. 133, Academic Press, Inc., Boston, MA, 1988.
  • [SS15] Emmanuel Schertzer and Florian Simatos, Height and contour processes of Crump-Mode-Jagers forests (i): general distribution and scaling limits in the case of short edges, 2015, arXiv:1506.03192.
  • [Wei07] Mathilde Weill, Regenerative real trees, Ann. Probab. 35 (2007), no. 6, 2091–2121.