Exponential convergence of products of random matrices: application to adaptive algorithms

Abstract

We introduce a novel methodology for analysing well known classes of adaptive algorithms. Combining recent developments concerning geometric ergodicity of stationary Markov processes and long existing results from the theory of Perturbations of Linear Operators we first study the behaviour and convergence properties of a class of products of random matrices, this is turn allows for the analysis of the first and second order statistics of adaptive algorithms without the need of any restrictive conditions imposed on the data (as essential boundedness). Efficient estimates of the convergence rate of adaptive algorithms during the initial transient phase are also presented. These estimates do not rely on the unrealistic Independence Assumption as it is commonly the case in existing literature. © 1998 John Wiley & Sons, Ltd.