The Annals of Probability

Nonconvergence to Unstable Points in Urn Models and Stochastic Approximations

Robin Pemantle

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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.

Article information

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

Dates
First available in Project Euclid: 19 April 2007

Permanent link to this document
https://projecteuclid.org/euclid.aop/1176990853

Digital Object Identifier
doi:10.1214/aop/1176990853

Mathematical Reviews number (MathSciNet)
MR1055428

Zentralblatt MATH identifier
0709.60054

JSTOR
links.jstor.org

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

Keywords
Reinforced random walk stochastic approximation unstable equilibrium urn model

Citation

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. https://projecteuclid.org/euclid.aop/1176990853


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