Convergence with probability one of stochastic approximation algorithms whose average is cooperative

We consider a stochastic approximation process where F : m m is a C 2 irreducible cooperative dissipative vector field, n n 0 is a sequence of positive numbers decreasing to 0 and U n n 0 a sequence of uniformly bounded m martingale differences. We show that under certain conditions on n and U n the sequence x n n 0 converges with probability one toward the equilibria set of the vector field F .