Regenerative tree growth: structural results and convergence
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 t...
| Main Authors: | , , |
|---|---|
| Formato: | Journal article |
| Idioma: | English |
| Publicado em: |
University of Washington
2014
|
| _version_ | 1826291015171964928 |
|---|---|
| author | Pitman, J Rizzolo, D Winkel, M |
| author_facet | Pitman, J Rizzolo, D Winkel, M |
| author_sort | Pitman, J |
| collection | OXFORD |
| description | 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 σ-finite dislocation measure κ on the set of partitions of N 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 κ, in addition to a regular variation condition. |
| first_indexed | 2024-03-07T02:53:01Z |
| format | Journal article |
| id | oxford-uuid:ae55c0e2-1e82-42fa-86fc-6c3ee0ef4392 |
| institution | University of Oxford |
| language | English |
| last_indexed | 2024-03-07T02:53:01Z |
| publishDate | 2014 |
| publisher | University of Washington |
| record_format | dspace |
| spelling | oxford-uuid:ae55c0e2-1e82-42fa-86fc-6c3ee0ef43922022-03-27T03:41:47ZRegenerative tree growth: structural results and convergenceJournal articlehttp://purl.org/coar/resource_type/c_dcae04bcuuid:ae55c0e2-1e82-42fa-86fc-6c3ee0ef4392EnglishSymplectic Elements at OxfordUniversity of Washington2014Pitman, JRizzolo, DWinkel, MWe 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 σ-finite dislocation measure κ on the set of partitions of N 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 κ, in addition to a regular variation condition. |
| spellingShingle | Pitman, J Rizzolo, D Winkel, M Regenerative tree growth: structural results and convergence |
| title | Regenerative tree growth: structural results and convergence |
| title_full | Regenerative tree growth: structural results and convergence |
| title_fullStr | Regenerative tree growth: structural results and convergence |
| title_full_unstemmed | Regenerative tree growth: structural results and convergence |
| title_short | Regenerative tree growth: structural results and convergence |
| title_sort | regenerative tree growth structural results and convergence |
| work_keys_str_mv | AT pitmanj regenerativetreegrowthstructuralresultsandconvergence AT rizzolod regenerativetreegrowthstructuralresultsandconvergence AT winkelm regenerativetreegrowthstructuralresultsandconvergence |