Limit theorems for the frontier of a one-dimensional branching diffusion
type: Item
instance of: scholarly article
title: Limit theorems for the frontier of a one-dimensional branching diffusion
zbMATH Open document ID: 0759.60087
author: Steven P. Lalley
author: Thomas M. Sellke
published in: The Annals of Probability
publication date: 1993-01-17T00:00:00Z
review text: Let \(R(t)\) be the position of the rightmost particle at time \(t\) in a time homogeneous and space nonhomogeneous one-dimensional branching diffusion process. Let \(\gamma(\alpha,t)\) be the \(\alpha\)-th quantile of \(R(t)\) of the process starting with a single particle at position 0. The authors show that \(\gamma(\alpha,t)\) is also a limiting \(\beta\)-th quantile of \(R(t-s)\) as \(t\to\infty\) of the process starting with a single particle at arbitrary position \(x\), where \(\beta=g(\alpha,S,X)\). In the last case the authors show that, after appropriate rescaling of space, the distribution of \(R(t)-t\) converges weakly to a nontrivial limiting distribution \(w_ x\) if the underlying diffusion is recurrent.
Mathematics Subject Classification ID: 60J80
Mathematics Subject Classification ID: 60G55
Mathematics Subject Classification ID: 60F05
zbMATH DE Number: 97397
zbMATH Keywords: travelling wave
zbMATH Keywords: extreme value distribution
zbMATH Keywords: branching diffusion process
zbMATH Keywords: rescaling of space
reviewed by: Valentin Topchii
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1214/aop/1176989693
OpenAlex ID: W1967869313
DOI: 10.1214/AOP/1176989693
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description: scientific article; zbMATH DE number 97397