One of the basic models in much of time series analysis is that of a weakly stationary process, a process whose mean value function is constant and whose covariance function depends only on the difference in the times at which the observations are made. The importance of this assumption is due to the fact that it implies that a Fourier (or harmonic) analysis of both the covariance function and the process itself can be carried out. These results will be developed in this chapter. As we shall later see, these basic probabilistic or structural results for these processes will motivate some of the statistical methods used in spectral analysis of data. The development will be carried out for discrete time parameter weakly stationary processes. However, there will be problems and occasional discussion suggesting analogous results and representations for continuous and or multidimensional time parameter processes. The results will be obtained for complex-valued weakly stationary processes.
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