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.
Publié le : 1990-04-14
Classification:
Reinforced random walk,
stochastic approximation,
unstable equilibrium,
urn model,
60G99,
62L20
@article{1176990853,
author = {Pemantle, Robin},
title = {Nonconvergence to Unstable Points in Urn Models and Stochastic Approximations},
journal = {Ann. Probab.},
volume = {18},
number = {4},
year = {1990},
pages = { 698-712},
language = {en},
url = {http://dml.mathdoc.fr/item/1176990853}
}
Pemantle, Robin. Nonconvergence to Unstable Points in Urn Models and Stochastic Approximations. Ann. Probab., Tome 18 (1990) no. 4, pp. 698-712. http://gdmltest.u-ga.fr/item/1176990853/