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New upper bounds on the norm of the error between a parameter vector obtained by an adaptive signal processing algorithm and the desired parameter vector are presented. The family of algorithms treated includes the Widrow-Hoff LMS algorithm. The results are applicable to heavily correlated training data satisfying very mild covariance decay-rate conditions. The main result is a new proof for the almost sure exponential convergence of matrix products which arise in the analysis of adaptive signal processing algorithms. This result includes an estimate for the convergence rate of such products.