• DOI:10.1214/AOP/1176990853
• Corpus ID: 122008076

Nonconvergence to Unstable Points in Urn Models and Stochastic Approximations

@article{Pemantle1990NonconvergenceTU,
  title={Nonconvergence to Unstable Points in Urn Models and Stochastic Approximations},
  author={Robin Pemantle},
  journal={Annals of Probability},
  year={1990},
  volume={18},
  pages={698-712}
}

• R. Pemantle
• Published 1 April 1990
• Mathematics
• Annals of Probability

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.