Nonconvergence to unstable points in urn models and stochastic approximations
type: Item
instance of: scholarly article
title: Nonconvergence to unstable points in urn models and stochastic approximations
zbMATH Open document ID: 0709.60054
author: Robin Pemantle
published in: The Annals of Probability
publication date: 1990-01-01T00:00:00Z
review text: Let F: \(R^ d\to R^ d\) be a vector field and consider a particle moving in discrete time whose position v(n) obeys the law \[ v(n+1)=v(n)+a\cdot F(v(n))+\xi_ n, \] where \(a_ n\) is an \({\mathcal F}_{\infty}\)-measurable random variable, \(\xi_ n\) is a random vector with \(E(\xi_ n| {\mathcal F}_ n)=0\), and \({\mathcal F}_ n\) is the \(\sigma\)-algebra of events up to time n. Let \(\Delta \subset R^ d\) be an open subset of an affine subspace in \(R^ d\), p be any point of \(\Delta\) with \(F(p)=0\), and p be a linearly unstable critical point, \(F\in C^ 2\). Under some conditions on \(a_ n,\xi_ n\) it is proved that \[ P\{v(n)\to p\}=0. \]
Mathematics Subject Classification ID: 60G99
Mathematics Subject Classification ID: 62L20
zbMATH DE Number: 4166212
zbMATH Keywords: unstable critical point
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1214/aop/1176990853
OpenAlex ID: W2015836240
DOI: 10.1214/AOP/1176990853
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description: scientific article; zbMATH DE number 4166212