Dimensions of supercritical branching processes in varying environments
type: Item
instance of: scholarly article
title: Dimensions of supercritical branching processes in varying environments
zbMATH Open document ID: 1067.60088
author: Narn-Rueih Shieh
author: Jinghu Yu
published in: Statistics \& Probability Letters
publication date: 2005-03-08T00:00:00Z
full work available at URL: 19.pdf
review text: Let \(Z(n)\) be a branching process in varying environment (BPVE) and \(u_n\) be the expected number of children of individuals of the \(n\)th generation. A BPVE is called uniformly supercritical if there are \(A>0\) and \(c>1\) such that \(\prod_{j=i}^{n+i-1}u_j\geq Ac^n,\;i,n\geq 1.\) Let \(\partial\Gamma\) be the boundary of a family tree \(\Gamma\) of a uniformly supercritical BPVE, and \(\dim_H\partial\Gamma,\) \(\overline{\dim}_B\partial\Gamma,\) \(\dim_P\partial\Gamma\) be the Hausdorff dimension, the upper box dimension and the packing dimension of \(\partial\Gamma,\) respectively. It is shown that \[ \dim_H\partial\Gamma=\liminf_{n\to\infty}n^{-1}\ln EZ(n),\quad \overline{\dim}_B\partial\Gamma=\dim_P\partial\Gamma=\limsup_{n\to\infty}n^{-1}\ln EZ(n). \]
reviewed by: Vladimir Vatutin
Mathematics Subject Classification ID: 60J80
Mathematics Subject Classification ID: 60G17
zbMATH DE Number: 2142300
zbMATH Keywords: Hausdorff dimension
zbMATH Keywords: upper box dimension
zbMATH Keywords: packing dimension
MaRDI profile type: Publication
OpenAlex ID: W2026304929
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DOI: 10.1016/J.SPL.2004.10.004
description: scientific article; zbMATH DE number 2142300