Discrete spectrum of nonstationary stochastic processes on groups

Vector-valued, asymptotically stationary stochastic processes on sigma-comp act locally compact abelian groups are studied. For such processes, we intr oduce a stationary spectral measure and show that it is discrete if and onl y if the asymptotically stationary covariance function is almost periodic. Using an "almost periodic Fourier transform" we recover the discrete part o f the spectral measure and construct a natural, consistent estimator for th e latter from samples of the process.