Correlation Theorem for Nonstationary Complex Stochastic Processes of a Real Variable

The object of this paper is to derive a general correlation theorem for a class of complex stochastic processes of a real argument. By means of this theorem, the correlation functions and the time-power spectral densities, which are defined by averaging the realizations of the processes and their running spectra, respectively, are related to each other by a pair of one-dimensional integral transformations. This theorem is reduced to corresponding theorems for other classes of stochastic processes which form subsets of the set of processes under consideration. The properties of the correlation functions and time-power spectral densities along with questions concerning conditions for the existence and usefulness of these concepts in scientific and engineering applications occupy a good portion of this paper.