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2014 Regenerative tree growth: structural results and convergence
Jim Pitman, Douglas Rizzolo, Matthias Winkel
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Electron. J. Probab. 19: 1-27 (2014). DOI: 10.1214/EJP.v19-3040

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.

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Jim Pitman. Douglas Rizzolo. Matthias Winkel. "Regenerative tree growth: structural results and convergence." Electron. J. Probab. 19 1 - 27, 2014. https://doi.org/10.1214/EJP.v19-3040

Information

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

zbMATH: 1304.60096
MathSciNet: MR3256870
Digital Object Identifier: 10.1214/EJP.v19-3040

Subjects:
Primary: 60J80

Keywords: Continuum random tree , fragmentation , Markov branching model , recursive random tree , regenerative composition , R-tree , self-similar tree , weighted R-tree

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