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A path of a stationary sequence of random variables $y_t$ does not have a discrete-time Fourier transform in the classical sense because it is not summable. This leads to considering the spectral representation theorem for stationary processes (a classical reference is Priestley's Spectral Analysis and Time Series).

However I believe such a path does admit a Fourier transform when seen as a tempered distribution $\sum_{- \infty}^\infty y_t \delta_t$ . What happens if you take this Fourier transform pointwise in $\omega$ ($\Omega$ the background probability space)?

  • Does this create a random variable in the space of tempered distributions?
  • Can you relate this candidate "random tempered distribution" to the spectral representation theorem or the power spectral density?

(I don't think discrete versus continuous time matters for my question.)

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