CONTRIBUTIONS TO EVOLUTIONARY SPECTRAL THEORY

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Abstract
The purpose of this paper is to discuss several fundamental issues in the theory of time‐dependent spectra for univariate and multivariate non‐stationary processes. The general framework is provided by Priestley's evolutionary spectral theory which is based on a family of stochastic integral representations. A particular spectral density function can be obtained from the Wold—Cramér decomposition, as illustrated by several examples. It is shown why the coherence is time invariant in the evolutionary theory and how the theory can be generalized so that the coherence becomes time dependent. Statistical estimation of the spectrum is also considered. An improved upper bound for the bias due to non‐stationarity is obtained which does not rely on the characteristic width of the process. The results obtained in the paper are illustrated using time series simulated from an evolving bivariate autoregressive moving‐average process of order (1, 1) with a highly time‐varying coherence.

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