Persistence of excitation conditions and the convergence of adaptive schemes

In the study of the behavior of adaptive filtering algorithms, persistence of excitation of the input process arises as a sufficient condition for convergence and, perhaps more importantly, for convergence rate of the parameter estimates. In this paper the underlying nature of the persistence requirement is presented and discussed, relating its normal specification in terms of moment conditions with covariance decays, etc., to sample path properties. Deterministic and stochastic persistence conditions and persistence measures are treated, as well as, persistence conditions for output-error, equation-error, and adaptive control schemes.