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Regenerative tree growth: structural results and convergence


 
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1. Title Title of document Regenerative tree growth: structural results and convergence
 
2. Creator Author's name, affiliation, country Jim Pitman; University of California at Berkeley
 
2. Creator Author's name, affiliation, country Douglas Rizzolo; University of Washington, Seattle
 
2. Creator Author's name, affiliation, country Matthias Winkel; University of Oxford
 
3. Subject Discipline(s)
 
3. Subject Keyword(s) regenerative composition, Markov branching model, fragmentation, self-similar tree, continuum random tree, R-tree, weighted R-tree, recursive random tree
 
3. Subject Subject classification 60J80
 
4. Description 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.
 
5. Publisher Organizing agency, location
 
6. Contributor Sponsor(s) NSF
 
7. Date (YYYY-MM-DD) 2014-08-15
 
8. Type Status & genre Peer-reviewed Article
 
8. Type Type
 
9. Format File format PDF
 
10. Identifier Uniform Resource Identifier http://ejp.ejpecp.org/article/view/3040
 
10. Identifier Digital Object Identifier 10.1214/EJP.v19-3040
 
11. Source Journal/conference title; vol., no. (year) Electronic Journal of Probability; Vol 19
 
12. Language English=en en
 
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