Magic Square of Real Spectral and Time Series Analysis with an Application to Moving Average Processes

Abstract

This paper is concerned with the spectral analysis of stochastic processes that are real-valued, one-dimensional, discrete-time, covariance-stationary, and which have a representation as a moving average (MA) process. In particular, we will review the meaning and interrelations of four fundamental quantities in the time and frequency domain, (1) the stochastic process itself (which includes filtered stochastic processes), (2) its autocovariance function, (3) the spectral representation of the stochastic process, and (4) the corresponding spectral distribution function, or if it exists, the spectral density function. These quantities will be viewed as forming the corners of a square (the “magic square of spectral and time series analysis”) with various connecting lines, which represent certain mathematical operations between them. To demonstrate the evaluation of these operations, we will discuss the example of a q-th order MA process.