Spectral representation of a periodic nonstationary random process

This paper deals with the periodic nonstationary process, the mean value and the correlation function of which are invariant under shift by a multiple of a certain period. The spectral representation is derived by making use of Loève's harmonizability theorem on a second-order nonstationary process. The process is represented as a sum of infinite stationary processes among which covariances exist. Each stationary process has a nonoverlapping frequency band of equal width, the center of which corresponds to a harmonic of the fundamental frequency determined by the period. The correlation function, dependent on two points, is represented in terms of a matrix-valued spectral density that is hermitian and nonnegative definite. The representations in other possible forms are also given. Finally some properties, special processes, and examples produced by a certain stationary random sequence are discussed.