Local dimensions of the branching measure on a Galton-Watson tree
type: Item
instance of: scholarly article
title: Local dimensions of the branching measure on a Galton-Watson tree
zbMATH Open document ID: 0986.60080
DOI: 10.1016/S0246-0203(00)01065-7
author: Quansheng Liu
published in: Annales de l'Institut Henri Poincaré. Probabilités et Statistiques
publication date: 2002-05-23T00:00:00Z
full work available at URL: http://www.numdam.org/item
full work available at URL: https://eudml.org/doc/77687
review text: Let \(\mu =\mu _{\omega}\) be the branching measure on the boundary \(\partial {\mathbf T}\) of a supercritical Galton-Watson tree \({\mathbf T}={\mathbf T}(\omega)\). Denote by \(\overline{d}(\mu, u)\) and \(\underline{d}(\mu, u)\) the upper and lower local dimensions of \(\mu \) at \(u\in \partial {\mathbf T}\), respectively. It is known that \(\overline{d}(\mu, u) = \underline{d}(\mu, u)=\log m\) for \(\mu\)-almost all \(u\in \partial {\mathbf T}\), where \(m=E{\mathbf N}\) is the expexted value of the offspring distribution \({\mathbf N}\). Assuming that \(E{\mathbf N}^{1+\delta}<\infty\) for some \(\delta >0\) the author finds necessary and sufficient conditions when the mentioned result holds for all \(u\in \partial {\mathbf T}\) and proves limit theorems concerning the local dimension of \(\mu\). He also finds the exact local dimensions of \(\mu\) at \(u\in \partial {\mathbf T}\) for \(\mu\)-almost all \(u\).
reviewed by: Vladimir Vatutin
Mathematics Subject Classification ID: 60J80
Mathematics Subject Classification ID: 28A78
Mathematics Subject Classification ID: 28A80
zbMATH DE Number: 1619322
zbMATH Keywords: Galton-Watson tree
zbMATH Keywords: branching measure
zbMATH Keywords: local dimension
zbMATH Keywords: exceptional points
zbMATH Keywords: uniform local dimensions
zbMATH Keywords: exact dimensions
MaRDI profile type: Publication
OpenAlex ID: W2028077977
description: scientific article; zbMATH DE number 1619322