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A discrete random walk $X_t$ starts today. We are asked what to prepare for the next years. Is this correct:

$ var(X_{2 years}) = 2 \: \: \int_{2\pi f}^{\pi} \:\: \phi(w) dw $

i.e. summing the power spectrum over the frequencies faster than 2 years?

I am aware the variance of Gaussian random walk is just $ n \sigma^2 $, however our real signal has more complex autocorrelation structure, and we have only an estimated power spectrum of it. Hence our methodological question.

For the continuous case I don't find the confirmation:

$ 2 \sigma^2 \int_{2 \pi f}^{\pi} \frac{1}{w^2} dw \neq \frac{\sigma^2}{f} $

Thanks for indicating us how we are misunderstanding the power spectrum concept.

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    $\begingroup$ It appears your question is entirely devoid of appropriate definitions of your notation. Please do rectify this. $\endgroup$ – wolfies Mar 8 '15 at 13:26
  • $\begingroup$ @wolfies: w = angular frequency, phi = power spectrum. Could you indicate which symbols are unclear? $\endgroup$ – user3817704 Mar 8 '15 at 13:35
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    $\begingroup$ Question is rather qualitative: Can the variance of a stochastic process Xt at a given point in the future be obtained from the power spectrum, or is that nonsense? $\endgroup$ – user3817704 Mar 8 '15 at 13:53
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    $\begingroup$ I don't think you can cut out the part of the spectrum for variance analysis. $\endgroup$ – Aksakal Mar 8 '15 at 14:10

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