Hereditary tree growth and Levy forests

STOCHASTIC PROCESSES AND THEIR APPLICATIONS

卷:129

Article

Duquesne, Thomas¹  Winkel, Matthias² 

[1] Sorbonne Univ, Fac Sci Pierre & Marie Curie, Lab Probabilites Stat & Modelisat, Boite Courrier 158,4 Pl Jussieu, F-75252 Paris, France

[2] Univ Oxford, Dept Stat, 1 South Pk Rd, Oxford OX1 3TG, England

关键词: Real tree;    Gromov-Hausdorff distance;    Galton-Watson tree;    Levy tree;    Leaf erasure;    Limit theorems;    Tightness;    Invariance principle;    Continuous-state branching process;   

DOI  :  10.1016/j.spa.2018.10.007

来源: Elsevier

【 摘 要 】

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 Levy forests. As a by-product of this convergence, we obtain a characterisation of the laws of Levy forests in terms of leaf-length erasure and we obtain invariance principles for discrete Galton-Watson trees, including the super-critical cases. (C) 2018 Elsevier B.V. All rights reserved.

【 授权许可】

Free