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

• A. M. Yaglom
Chapter

Abstract

The well-known problem of least squares prediction of the stationary stochastic process x (t) is the problem of finding the functional $$\tilde{x}\text{ }\left( \tau \right)$$ 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 $$\tilde{x}\text{ }\left( \tau \right)$$ it is important to know the mean square error of the prediction
$${{\sigma }^{2}}\left( \tau \right)\text{ = E}{{\left| x\left( \tau \right)-\tilde{x}\left( \tau \right) \right|}^{2}},$$
(1)
or, what is equivalent, the correlation coefficient
$$\ell \left( \tau \right)=\frac{Ex\left( \tau \right)\tilde{x}\left( \tau \right)}{{{\left\{ E{{[x\left( \tau \right)]}^{2}}E{{[\tilde{x}\left( \tau \right)]}^{2}} \right\}}^{\frac{1}{2}}}}=\sqrt{1-\frac{{{\sigma }^{2}}\left( \tau \right)}{E{{[x\left( \tau \right)]}^{2}}}}$$
(2)
between x (τ) and $$\tilde{x}\left( \tau \right)$$ [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 $$\tilde{x}\left( \tau \right)$$ 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).

Keywords

Spectral Density Gaussian Process Spectral Function Canonical Correlation Canonical Variable
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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