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April, 1990 Nonconvergence to Unstable Points in Urn Models and Stochastic Approximations
Robin Pemantle
Ann. Probab. 18(2): 698-712 (April, 1990). DOI: 10.1214/aop/1176990853
Abstract

A particle in $\mathbf{R}^d$ moves in discrete time. The size of the $n$th step is of order $1/n$ and when the particle is at a position $\mathbf{v}$ the expectation of the next step is in the direction $\mathbf{F}(\mathbf{v})$ for some fixed vector function $\mathbf{F}$ of class $C^2$. It is well known that the only possible points $\mathbf{p}$ where $\mathbf{v}(n)$ may converge are those satisfying $\mathbf{F}(\mathbf{p}) = \mathbf{0}$. 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 $(d/dt)\mathbf{v}(t) = \mathbf{{F}({v}}(t))$. This generalizes several known results that say convergence is impossible at a repelling node of the flow.

Pemantle: Nonconvergence to Unstable Points in Urn Models and Stochastic Approximations
Copyright © 1990 Institute of Mathematical Statistics
Robin Pemantle "Nonconvergence to Unstable Points in Urn Models and Stochastic Approximations," The Annals of Probability 18(2), 698-712, (April, 1990). https://doi.org/10.1214/aop/1176990853
Published: April, 1990
Vol.18 • No. 2 • April, 1990
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