Hereditary tree growth and Lévy forests
We introduce the notion of a hereditary property for rooted real trees and we also consider reduction of trees by a given hereditary property. Leaf-length erasure, also called trimming, is included as a special case of hereditary reduction. We only consider the metric structure of trees, and our fra...
| Main Authors: | , |
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| Format: | Journal article |
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Elsevier
2018
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| _version_ | 1797060416045580288 |
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| author | Duquesne, T Winkel, M |
| author_facet | Duquesne, T Winkel, M |
| author_sort | Duquesne, T |
| collection | OXFORD |
| description | We introduce the notion of a hereditary property for rooted real trees and we also consider reduction of trees by a given hereditary property. Leaf-length erasure, also called trimming, is included as a special case of hereditary reduction. We only consider the metric structure of trees, and our framework is the space of pointed isometry classes of locally compact rooted real trees equipped with the Gromov–Hausdorff distance. We discuss general tightness criteria in and limit theorems for growing families of trees. We apply these results to Galton–Watson trees with exponentially distributed edge lengths. This class is preserved by hereditary reduction. Then we consider families of such Galton–Watson trees that are consistent under hereditary reduction and that we call growth processes. We prove that the associated families of offspring distributions are completely characterised by the branching mechanism of a continuous-state branching process. We also prove that such growth processes converge to Lévy forests. As a by-product of this convergence, we obtain a characterisation of the laws of Lévy forests in terms of leaf-length erasure and we obtain invariance principles for discrete Galton–Watson trees, including the super-critical cases. |
| first_indexed | 2024-03-06T20:16:48Z |
| format | Journal article |
| id | oxford-uuid:2c6672fa-3f97-4954-ba4c-460bf0870299 |
| institution | University of Oxford |
| last_indexed | 2024-03-06T20:16:48Z |
| publishDate | 2018 |
| publisher | Elsevier |
| record_format | dspace |
| spelling | oxford-uuid:2c6672fa-3f97-4954-ba4c-460bf08702992022-03-26T12:36:58ZHereditary tree growth and Lévy forestsJournal articlehttp://purl.org/coar/resource_type/c_dcae04bcuuid:2c6672fa-3f97-4954-ba4c-460bf0870299Symplectic Elements at OxfordElsevier2018Duquesne, TWinkel, MWe introduce the notion of a hereditary property for rooted real trees and we also consider reduction of trees by a given hereditary property. Leaf-length erasure, also called trimming, is included as a special case of hereditary reduction. We only consider the metric structure of trees, and our framework is the space of pointed isometry classes of locally compact rooted real trees equipped with the Gromov–Hausdorff distance. We discuss general tightness criteria in and limit theorems for growing families of trees. We apply these results to Galton–Watson trees with exponentially distributed edge lengths. This class is preserved by hereditary reduction. Then we consider families of such Galton–Watson trees that are consistent under hereditary reduction and that we call growth processes. We prove that the associated families of offspring distributions are completely characterised by the branching mechanism of a continuous-state branching process. We also prove that such growth processes converge to Lévy forests. As a by-product of this convergence, we obtain a characterisation of the laws of Lévy forests in terms of leaf-length erasure and we obtain invariance principles for discrete Galton–Watson trees, including the super-critical cases. |
| spellingShingle | Duquesne, T Winkel, M Hereditary tree growth and Lévy forests |
| title | Hereditary tree growth and Lévy forests |
| title_full | Hereditary tree growth and Lévy forests |
| title_fullStr | Hereditary tree growth and Lévy forests |
| title_full_unstemmed | Hereditary tree growth and Lévy forests |
| title_short | Hereditary tree growth and Lévy forests |
| title_sort | hereditary tree growth and levy forests |
| work_keys_str_mv | AT duquesnet hereditarytreegrowthandlevyforests AT winkelm hereditarytreegrowthandlevyforests |