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Hereditary tree growth and Lévy forests

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  • Duquesne, Thomas
  • Winkel, Matthias

Abstract

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 T of pointed isometry classes of locally compact rooted real trees equipped with the Gromov–Hausdorff distance. We discuss general tightness criteria in T 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.

Suggested Citation

  • Duquesne, Thomas & Winkel, Matthias, 2019. "Hereditary tree growth and Lévy forests," Stochastic Processes and their Applications, Elsevier, vol. 129(10), pages 3690-3747.
    • Handle: RePEc:eee:spapps:v:129:y:2019:i:10:p:3690-3747
    DOI: 10.1016/j.spa.2018.10.007
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    References listed on IDEAS

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    1. Abraham, Romain & Delmas, Jean-François, 2009. "Williams' decomposition of the Lévy continuum random tree and simultaneous extinction probability for populations with neutral mutations," Stochastic Processes and their Applications, Elsevier, vol. 119(4), pages 1124-1143, April.
    2. Helland, Inge S., 1978. "Continuity of a class of random time transformations," Stochastic Processes and their Applications, Elsevier, vol. 7(1), pages 79-99, March.
    3. Grimvall, Anders, 1973. "On the transition from a Markov chain to a continuous time process," Stochastic Processes and their Applications, Elsevier, vol. 1(4), pages 335-368, October.
    4. Bingham, N. H., 1976. "Continuous branching processes and spectral positivity," Stochastic Processes and their Applications, Elsevier, vol. 4(3), pages 217-242, August.
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