A LAW OF LARGE NUMBERS FOR RESCALED RANDOM DIFFERENCE EQUATIONS

We study the behavior of stochastic processes defined as an iterated function system with initial value X0 = x0 and a stationary ergodic input signal (Un)n≥0 for small values of the parameter a. We obtain almost sure convergence of the path to the solution of the corresponding deterministic dynamical system defined by , where F(y) = E(f(y,U)). The results have applications in the study of neural network learning algorithms.