This paper investigates the dynamical properties of a class of urn processes
and recursive stochastic algorithms with constant gain which arise frequently
in control, pattern recognition, learning theory, and elsewhere.
It is shown that, under suitable conditions, invariant measures of the
process tend to concentrate on the Birkhoff center of irreducible (i.e.
chain transitive) attractors of some vector field $F: {\Bbb R}^d \rightarrow
{\Bbb R}^d$ obtained by averaging. Applications are given to simple
situations including the cases where $F$ is Axiom A or Morse–Smale, $F$
is
gradient-like, $F$ is a planar vector field, $F$ has finitely many alpha and
omega limit sets.