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Mathematical Statistics and Stochastic Processes by Denis Bosq

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Chapter 10

Weakly Stationary Discrete-Time Processes

10.1. Autocovariance and spectral density

1) Let images be a real, weakly stationary process with basis space images. The Xt then belong to the Hilbert space images and Cov(Xs, Xt) is the scalar product of Xs with Xt. Let us set:

images

Then images and the relation Cov(Xs, Xt) = γts 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 images minimizes the quadratic error:

images

where the solution is the orthogonal projection of Xn+1 onto the vector space generated ...

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