Fixed Gain Off-line Estimators of ARMA Parameters

Let b ` N denote the estimator of the ARMA parameter vector ` using a fixed gain off-line prediction error method with gain or forgetting rate . We show that under certain conditions b ` N \Gamma ` is an L-mixing process and can be decomposed as the sum of an explicitly given L-mixing process of the order of magnitude O( 1=2 ) and of a residual term of the order of magnitude O() + O((1 \Gamma ) N ). Here the order of magnitude is measured as the Lq(\Omega ; F ; P ) norm for any q 1 where(\Omega ; F ; P ) is the underlying probability space. The result of the paper is directly applicable to fixed gain recursive estimators of AR models. The result of this paper has been applied in the theory of stochastic complexity. Key words: ARMA-processes, prediction error estimation, forgetting, strong approximation, L-mixing processes AMS Subject Classifications: 60H10, 62L20, 93E12 1 Introduction It has been shown in [5] that the estimation error of the parameters of an ARMA ..