Nonconvergence to unstable points in urn models and stochastic approximations
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 \[ P\{v(n)\to p\}=0. \]
scientific article; zbMATH DE number 4166212