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Sufficient conditions are presented for establishing "desirable" convergence properties of commonly used adaptive signal processing algorithms which use correlated training data. The family of algoriths considered includes the Widrow LMS algorithm. Desirable properties include, e.g., an asymptotic bound on the mean-square error between the parameter vector trained by the adaptive algorithm and the optimal solution. This asymptotic bound should decrease with decreasing step size. The results contained in this paper illustrate the trade-offs involved in choosing the step size to achieve an acceptable convergence rate as well as an acceptable steady state error. The sufficient conditions include bounded data and easily verified covariance decay rate conditions.