A conditional limit theorem for tree-indexed random walk
type: Item
instance of: scholarly article
title: A conditional limit theorem for tree-indexed random walk
zbMATH Open document ID: 1093.60061
published in: Stochastic Processes and their Applications
publication date: 2006-04-28T00:00:00Z
review text: Let \(\mathcal{T}\) be a Galton-Watson tree with critical aperiodic offspring distribution with analytic characteristic function and conditioned to have exactly \(n\) vertices. Let \(Y_e\) be i.i.d. r.v.'s with symmetric distribution \(\gamma\) such that as \(x\to \infty\), \(\gamma(x,\infty)= o(x^{-4})\). It is assumed that these \(Y_e\) are associated to different edges of the tree, and the spatial position \(U_v\) of a vertex \(v\) is obtained by summing the displacements \(Y_e\) along the ray from the root to the \(v\). The resulting object \((\mathcal{T}, \{U_v, v\in \mathcal{T}\})\) is called discrete spatial tree. The main result of the paper states that the scaling limit of the pair consisting of the contour function and the spatial contour function of the discrete spatial trees conditioned to have nonnegative spatial positions is a conditioned Brownian snake with initial point \(0\). Also the author applies his main result to give a short proof of some asymptotics for random quadrangulations which were obtained by \textit{P. Chassaing} and \textit{G. Schaeffer} [Probab. Theory Relat. Fields 128, No. 2, 161-212 (2004; Zbl 1041.60008)].
reviewed by: Alexander Iksanov
Mathematics Subject Classification ID: 60J80
zbMATH DE Number: 5021136
zbMATH Keywords: Galton-Watson tree
zbMATH Keywords: spatial tree
zbMATH Keywords: conditioned tree
zbMATH Keywords: conditioned Brownian snake
zbMATH Keywords: random quadrangulations
zbMATH Keywords: invariance principle
zbMATH Keywords: ISE
zbMATH Keywords: well-labelled tree
author: J.-F. Le Gall
MaRDI profile type: Publication
full work available at URL: j.spa.2005.11.008
OpenAlex ID: W2040766831
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DOI: 10.1016/J.SPA.2005.11.008
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description: scientific article