Electronic Journal of Probability

Regenerative tree growth: structural results and convergence

Jim Pitman, Douglas Rizzolo, and Matthias Winkel

Full-text: Open access

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

Permanent link to this document
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

Subjects
Primary: 60J80: Branching processes (Galton-Watson, birth-and-death, etc.)

Keywords
regenerative composition Markov branching model fragmentation self-similar tree continuum random tree R-tree weighted R-tree recursive random tree

Rights
This work is licensed under a Creative Commons Attribution 3.0 License.

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


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