## The Annals of Probability

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

### Nonconvergence to Unstable Points in Urn Models and Stochastic Approximations

#### 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