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Stationary 2nd-Order Processes

  • John Lamperti
Part of the Applied Mathematical Sciences book series (AMS, volume 23)

Abstract

This chapter begins the more particular theory of stationary 2nd-order random processes, considered from the view-point of correlation theory. In other words, we will study processes which are “stationary in the wide sense” (page 7) and build a theory based on their covariance functions \({\rm K(s) = E(X}_{{\rm t + s}} \overline {\rm X} _{\rm t} )\) alone. This theory has the flavor of Hilbert space and Fourier analysis, and readers who are familiar with the “spectral theorem” for unitary operators on a Hilbert space will recognize that this theorem is behind the “spectral representation” of a stationary process to be derived below. No advance knowledge of spectral theory is needed, however, and in fact the probabilistic setting can provide an easy and well-motivated introduction to this area of functional analysis.

Keywords

Covariance Function Spectral Measure Spectral Representation Trigonometric Polynomial Spectral Form 
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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Copyright information

© Springer-Verlag, New York Inc. 1977

Authors and Affiliations

  • John Lamperti
    • 1
  1. 1.Department of MathematicsDartmouth CollegeHanoverUSA

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