Almost-sure sample stability of the continuous-time LMS algorithm

Consideration is given to the stability properties of the conventional continuous-time LMS algorithm. The algorithm is investigated for the case of stationary ergodic inputs, 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 connection between persistency of excitation and positive-definite autocorrelation in stationary ergodic vector waveforms