Stationary Gaussian Processes Satisfying the Strong Mixing Condition and Best Predictable Functionals

Abstract

The well-known problem of least squares prediction of the stationary stochastic process x (t) is the problem of finding the functional $\widetilde{x}(\tau )$ of the values x (t), t ≦ 0, which is the least squares approximation of the “future” value of the process x (τ), τ > 0. In addition to the functional $\widetilde{x}(\tau )$ it is important to know the mean square error of the prediction

$${{\sigma }^{2}}(\tau )=E|x(\tau )-\widetilde{x}(\tau ){{|}^{2}},$$

(1)

or, what is equivalent, the correlation coefficient

$$\rho (\tau )=\frac{Ex(\tau )\widetilde{x}(\tau )}{{{\{E{{[x(\tau )]}^{2}}E{{[\widetilde{x}(\tau )]}^{2}}\}}^{\frac{1}{2}}}}=\sqrt{1-\frac{{{\sigma }^{2}}(\tau )}{E{{[x(\tau )]}^{2}}}}$$

(2)

between x (τ) and $\widetilde{x}(\tau )$ [here and later we can consider without loss of generality only the processes x (t) with Ex (t) = 0]. If the process x (t) is Gaussian, the least squares approximation $\widetilde{x}(\tau )$ is linear; therefore, we can say that the problem of linear least squares prediction of the stationary process x (t) is the wide sense version of the general problem of least squares prediction (see Doob [1], Chapter II Section 3).