Almost-sure convergence of the continuous-time LMS algorithm

The authors consider the stability properties of the conventional continuous-time least mean square algorithm. The algorithm for the case of stationary ergodic inputs is investigated and a necessary and sufficient condition for exponential almost-sure convergence is presented. This condition is shown to be less restrictive than the well-known persistency of excitation condition. Also, the authors point out and clarify an apparently common error regarding the connections between persistency of excitation and positive definite autocorrelation in stationary ergodic vector waveforms