Periodogram Analysis and Continuous Spectra

Abstract

Abstract It has been realized for some time that the classical method of searching for periodicities in time series, the so-called periodogram analysis of the series, is useless in many cases. An alternative approach by way of the autoregressive or autocorrelative structure of the series was considered in a pioneering paper by Yule (1927), and more recently by other writers (see especially M. G. Kendall (1946)). A fuller understanding of the scope and relation of these methods is obtainable from the mathematical theory of stationary random or stochastic processes developed in the last twenty years by Wiener (1930), Khintchine (1934) and others (for an account of this theory see Lévy (1948), especially chapter IV; for further reviews of the general theory of stochastic processes see also Cramér (1947), Moyal (1949)). By stationary is meant, roughly speaking, that the series is oscillating or fluctuating about a constant mean; more precisely, that its distributional properties in the stochastic sense do not depend on the absolute value of the time (note that a simple harmonic series is not stationary in this sense unless its initial phase is random). For many purposes in the correlation and harmonic theory of these processes it is sufficient to assume stationarity ‘to the second order’, that is, in addition to the mathematical expectation or stochastic average E{X(t)} =constant= 0, say, for the value of the series X at time t, the autocovariance function