A Law of Large Numbers for Rescaled Random Difference Equations∗

Abstract: We study the behaviour of stochastic processes defined as an iterated function system Xn+1 = Xn + af(Xn, Un+1) 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 ẏ = F (y), where F (y) = E(f(y, U)). The results have applications in the study of neural network learning algorithms.