Fourier Analysis of Periodic Weakly Stationary Processes: A Note on Slutsky's Observation

The periodic behavior of a specific weakly stationary stochastic process (w.s.p.) is examined from a viewpoint of classical Fourier analysis. (1) A w.s.p. has a spectral measure which is absolutely continuous with respect to the Lebesgue measure if and only if it is a moving average of a white noise. (2) A periodic or almost periodic w.s.p. must have a "discrete" spectral measure. Combining these two, we can conclude that any moving average of a white noise can neither be periodic nor almost periodic. However any w.s.p. can be approximated by a sequence of almost periodic w.s.p.'s in some specific sense.