Open AccessA Law of Large Numbers for Rescaled Random Difference Equations
Author(s) -
Robert Burton,
Herold Dehling,
Uwe Rösler
Publication year - 2003
Publication title -
stochastics and dynamics
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.566
H-Index - 26
eISSN - 1793-6799
pISSN - 0219-4937
DOI - 10.1142/s0219493703000760
Subject(s) - mathematics , ergodic theory , law of the iterated logarithm , law of large numbers , iterated function system , iterated function , dynamical systems theory , function (biology) , dynamical system (definition) , path (computing) , convergence (economics) , combinatorics , mathematical analysis , discrete mathematics , random variable , attractor , statistics , physics , quantum mechanics , logarithm , evolutionary biology , economics , biology , economic growth , computer science , programming language
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.
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