Nonconvergence to Unstable Points in Urn Models and Stochastic Approximations

A particle in Rᵈ moves in discrete time. The size of the nth step is of order $1/n$ and when the particle is at a position v the expectation of the next step is in the direction F(v) for some fixed vector function F of class C². It is well known that the only possible points p where v(n) may converge are those satisfying F(p) = 0. This paper proves that convergence to some of these points is in fact impossible as long as the "noise"--the difference between each step and its expectation--is sufficiently omnidirectional. The points where convergence is impossible are the unstable critical points for the autonomous flow (d/dt)v(t) = F(v(t)). This generalizes several known results that say convergence is impossible at a repelling node of the flow.