The Spectral Representation [⋆]
Tucker McElroy, Dimitris N. Politis · 2019
This chapter departs from the prior autoregressive moving average framework, providing general frequency-domain representations for a stationary time series that hold true even when the spectral density might not exist. The Herglotz Theorem provides a converse assertion that is true in a general setting – even when the spectral density does not exist. Talking about spectral density and spectral distribution is in analogy to the notions of probability density and probability distribution of a random variable. The Herglotz Theorem indicates that all autocovariance functions correspond to a spectral distribution function, which is bounded, non-decreasing, and right continuous. The chapter defines the spectral representation of a discrete-time stationary stochastic process. It invokes the theory of stochastic processes with orthogonal increments, in Appendix E.