Consistency of adaptive estimators on the basis of correlated observations (Q1968879)

Suppose that a set of observations has the following structure \[ y_t= \sum^n_{k=1} c_k\varphi_k(t)+ \xi_t, \] where \(t= 1,2,\dots, y_t\) is an observation of the output variable, \(\varphi_k(t)\) are known values, \(\xi_t\) is the usual additive noise, and \(c_k\) are unknown parameters that must be estimated in the process of identification. The authors extend the least squares method to the case of dependent observations. They present the form of the adaptive estimator constructed by the least squares method for the case when the components of the noise vector are correlated. The correlation between observations sometimes leads to the absence of the consistent estimator of the parameters \(c_k\). The authors consider the case when \(c_k= c\) is an unknown mathematical expectation, \(y_t= c+\xi_t\). They formulate sufficient conditions for the consistency of the estimator constructed by the least squares method.