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Exponential convergence of adaptive algorithms

We introduce a novel method for analyzing a well known class of adaptive algorithms. By combining developments from the theory of Markov processes and long existing results from the theory of perturbations of linear operators we study first the behavior and convergence properties of a class of products of random matrices. This in turn allows for the analysis of the first and second order statistics of the adaptive algorithms yielding estimates for the exponential rate of convergence and the covariance matrix of the estimation error.