Poisson point process limits in size-biased Galton-Watson trees
type: Item
instance of: scholarly article
title: Poisson point process limits in size-biased Galton-Watson trees
zbMATH Open document ID: 0963.60084
author: J. Geiger
published in: Electronic Journal of Probability
publication date: 2001-01-09T00:00:00Z
full work available at URL: https://eudml.org/doc/121036
full work available at URL: paper17.abs.html
review text: Let \(Z_t\), \(t\geq 0,\) be a continuous-time critical binary Galton-Watson branching process starting with a single founding ancestor, and let \(\widehat{T}_t\) be the tree of the process size-biased according to the number of particles at time \(t.\) By keeping track of 1) the times when different families grow out of the distinguished line of descent of a particle picked at random among existing at time \(t,\) 2) the sizes of the respective families at time \(t,\) the author constructs a point process in a time-size plane and studies its properties for the cases of single-type and some multi-type continuous-time Galton-Watson processes. As a corollary, he shows in particular, that the number of edges of height \(t\) in \(\widehat{T}_t\) has a gamma limit law with shape parameter 2.
Mathematics Subject Classification ID: 60J80
Mathematics Subject Classification ID: 60G55
zbMATH DE Number: 1549920
zbMATH Keywords: Galton-Watson process
zbMATH Keywords: random tree
zbMATH Keywords: limit laws
reviewed by: Vladimir Vatutin
MaRDI profile type: Publication
DOI: 10.1214/EJP.V5-73
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description: scientific article; zbMATH DE number 1549920