# A representation theorem for the error of recursive estimators

Gerencsér, László
(2006)
*A representation theorem for the error of recursive estimators.*
In: CDC 2006. 45th IEEE conference on decision and control. San Diego, 2006..

## Abstract

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 $q ge 1$, the $L_q$-norms of the so-called residual terms are shown to tend to zero with rate $N^{-1/2-varepsilon}$ with some $varepsilon > 0$. This is a significant extension of previous results for the recursive prediction error or RPE estimator of ARMA processes given in [L.~Gerencs'er, {it 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.

Item Type: | Conference or Workshop Item (Paper) |
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Uncontrolled Keywords: | adaptive prediction, stochastic complexity, recursive estimation, $L$-mixing processes, asymptotic covariance, stochastic adaptive control |

Subjects: | Q Science > QA Mathematics and Computer Science > QA75 Electronic computers. Computer science / számítástechnika, számítógéptudomány |

Depositing User: | Eszter Nagy |

Date Deposited: | 11 Dec 2012 15:26 |

Last Modified: | 11 Dec 2012 15:26 |

URI: | http://eprints.sztaki.hu/id/eprint/4503 |

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