Convergence with probability one of stochastic approximation algorithms whose average is cooperative - Benaim, Michel (My University, 2000)
We consider a stochastic approximation process Xn+1 - x(n) = Yn+1 (F(x(n)) + Un+1) where F : R-m --> R-m is a C-2 irreducible cooperative dissipative vector field, {y(n)}(n greater than or equal to 0) is a sequence of positive numbers decreasing to 0 and {U-n}(n greater than or equal to 0) a sequence of uniformly bounded R-m martingale differences. We show that under certain conditions on {y(n)} and {U-n} the sequence {x(n)}(n greater than or equal to 0) converges with probability one toward the equilibria set of the vector field F.
Keywords: DYNAMICAL-SYSTEMS; MAPS