We consider stochastic approximation algorithms with constant step size whose average ordinary differential equation (ODE) is cooperative and irreducible. We show that, under mild conditions on the noise process, invariant measures and empirical occupations measures of the process weakly converge (as the time goes to infinity and the step size goes to zero) toward measures which are supported by stable equilibria of the ODE. These results are applied to analyzing the long-term behavior of a class of learning processes arising in game theory.
"Stochastic approximation algorithms with constant step size whose average is cooperative." Ann. Appl. Probab. 9 (1) 216 - 241, February 1999. https://doi.org/10.1214/aoap/1029962603