A Law of Large Numbers for Rescaled Random Difference Equations
Robert Main Burton, Herold Dehling, Uwe Rösler · Stochastics and Dynamics · 2003
We study the behavior of stochastic processes defined as an iterated function system [Formula: see text] with initial value X0= x0and a stationary ergodic input signal (Un)n≥0for 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 [Formula: see text], where F(y) = E(f(y,U)). The results have applications in the study of neural network learning algorithms.