The Annals of Probability

Nonconvergence to Unstable Points in Urn Models and Stochastic Approximations

Robin Pemantle

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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.

Article information

Ann. Probab., Volume 18, Number 2 (1990), 698-712.

First available in Project Euclid: 19 April 2007

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier


Primary: 60G99: None of the above, but in this section
Secondary: 62L20: Stochastic approximation

Reinforced random walk stochastic approximation unstable equilibrium urn model


Pemantle, Robin. Nonconvergence to Unstable Points in Urn Models and Stochastic Approximations. Ann. Probab. 18 (1990), no. 2, 698--712. doi:10.1214/aop/1176990853.

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