Asymptotic Properties of the LS-estimator of a Gaussian Autoregressive Process by an Averaging Method

Abstract

This article deals with the problem of parameter estimation of a continuous-time p-dimensional Gaussian autoregressive process. In the stable case, we combine averaging and weighting methods to establish, for the weighted LS-estimator (least-squares estimator) of θ, an almost-sure central limit theorem (ASCLT), a quadratic strong law (QSL) and a central limit theorem (CLT) associated to the QSL with arithmetic convergence rates. In the unstable case, we establish for the LS-estimator of θ, the same optimal asymptotic properties with arithmetic convergence rates. In both cases, using the QSL property, we construct a new estimator of the noise covariance σ2 and we specify its weak convergence rates.