Stationary 2nd-Order Processes

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.