1) Let be a real, weakly stationary process with basis space . The Xt then belong to the Hilbert space and Cov(Xs, Xt) is the scalar product of Xs with Xt. Let us set:
Then and the relation Cov(Xs, Xt) = γt−s shows that (γt) completely determines the covariance of (Xt). (γt) is said to be the autocovariance of (Xt). The sequence (γt) is very important, since it provides all of the information about the linear correlation between the random variables Xt.
2) For example, if we seek to determine the linear regression of Xn+1 onto X1, …, Xn, that is, the random variable of the form minimizes the quadratic error:
where the solution is the orthogonal projection of Xn+1 onto the vector space generated ...