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

  • Toru MaruyamaEmail author
Mini Course
Part of the Advances in Mathematical Economics book series (MATHECON, volume 20)


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.


Weakly stationary process Periodicity Almost periodicity Spectral measure 


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© Springer Science+Business Media Singapore 2016

Authors and Affiliations

  1. 1.Keio UniversityTokyoJapan

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