Recursive algorithms, urn processes andchaining number of chain recurrent sets

Abstract

This paper investigates the dynamical properties of a class of urn processesand recursive stochastic algorithms with constant gain which arise frequentlyin control, pattern recognition, learning theory, and elsewhere.

It is shown that, under suitable conditions, invariant measures of theprocess 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 simplesituations including the cases where $F$ is Axiom A or Morse–Smale, $F$isgradient-like, $F$ is a planar vector field, $F$ has finitely many alpha andomega limit sets.