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

  • A. M. Yaglom


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
$${{\sigma }^{2}}\left( \tau \right)\text{ = E}{{\left| x\left( \tau \right)-\tilde{x}\left( \tau \right) \right|}^{2}},$$
or, what is equivalent, the correlation coefficient
$$\ell \left( \tau \right)=\frac{Ex\left( \tau \right)\tilde{x}\left( \tau \right)}{{{\left\{ E{{[x\left( \tau \right)]}^{2}}E{{[\tilde{x}\left( \tau \right)]}^{2}} \right\}}^{\frac{1}{2}}}}=\sqrt{1-\frac{{{\sigma }^{2}}\left( \tau \right)}{E{{[x\left( \tau \right)]}^{2}}}}$$
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).


Manifold Covariance Dition 


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© Springer-Verlag Berlin · Heidelberg 1965

Authors and Affiliations

  • A. M. Yaglom
    • 1
  1. 1.Academy of Sciences of the USSRRussia

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