Spectral representation and ergodicity of a class of nonstationary stochastic processes

Spectral analysis of stationary processes has played an essential role in the development of Time Series. Nonstationary processes are less well understood however, although their consideration seems more realistic for many important applications. A summary of existing theory for stationary processes as well as a survey of the various classes of nonstationary processes that admit spectra will introduce this thesis. By combining and extending the ideas from the Wiener approach and the Khintchine-Kolmogorov-Cramer approach to stationary time series, a spectral representation and a mean ergodic theorem will be obtained under very mild conditions for a large class of nonstationary time series - the class of asymptotically stationary processes. Following these methods, a spectral representation for the class of asymptotically stationary random fields will be derived. The class of almost harmonizable random fields will be shown to be contained in the class of asymptotically stationary random fields. As an example of asymptotically stationary random fields, we consider the class of periodically correlated random fields. We give an upper bound for the covariance function of this class. A simulation is carried out to evaluate the accuracy of this approximation for the random field generated from a nonstationary random step function first defined by Wiener (1930). This result is useful in characterizing and distinguishing periodic textures. A spectral representation is derived for a class of random fields which are bounded in mean square almost surely. A characterization of the corresponding spectral measure in the representation is obtained based on Beurling's duality theorem and generalized Fourier transforms. A representation for the covariance function is also defined for this large class which contains the class of asymptotically stationary random fields