Fourier Analysis of Time Series

Abstract

Discrete-time wide-sense stationary stochastic processes, also called time series, arise from discrete-time measurements (sampling) of random functions. A particularly mathematically tractable class of such processes consists of the so-called moving averages and auto-regressive (and more generally, arma) time series. This chapter begins with the general theory of wss discrete-time stochastic processes (which essentially reproduces that of wss continuous-time stochastic processes) and then gives the representation theory of arma processes, together with their prediction theory. The last section is concerned with the realization problem: what models fit a given finite segment of autocorrelation function of a time series? The corresponding theory is the basis of parametric spectral analysis.