A Representation Theorem for the Error of Recursive Estimators

The objective of this paper is to present advanced and less known techniques for the analysis of performance degradation due to statistical uncertainty for a wide class of linear stochastic systems in a rigorous and concise manner. The main technical advance of the present paper is a strong approximation theorem for the Djereveckii-Fradkov-Ljung (DFL) scheme with enforced boundedness, in which, for any q ges 1, the L_q-norms of the so-called residual terms are shown to tend to zero with rate N^{-frac12-epsiv} with some epsiv > 0. This is a significant extension of previous results for the recursive prediction error or RPE estimator of ARMA processes given in [L. Gerencser, Systems Control Lett., 21 (1993), pp. 347-351. Two useful corollaries will be presented. In the first a standard transform of the estimation-error process will be shown to be L-mixing. In the second the asymptotic covariance matrix of the estimator will be given. An application to the minimum-variance self-tuning regulator for ARMAX systems will be described