ABSTRACT
This chapter deals with the spectral representation of weakly stationary processes – stationary in the sense that the mean is constant and the covariance C(x(s),x(t)) only depends on the time difference s− t. For real-valued Gaussian processes, the mean and covariance function determine all finitedimensional distributions, and hence the entire process distribution. However, the spectral representation requires complex-valued processes, and then it is necessary that one specifies also the correlation structure between the real and the imaginary part of the process. We therefore start with a summary of the basic properties of complex-valued processes, in general, and in the Gaussian case, before we introduce and prove the spectral representation of the covariance function, and subsequently, of the process itself. We remind the reader of the classical memoirs by S.O. Rice [100], which can be recommended to anyone with the slightest historical interest. That work also contains many old references.
