G. M. Raymond, D. B. Percival and J. B. Bassingthwaighte (2003), `The Spectra and Periodograms of Anti-correlated Discrete Fractional Gaussian Noise,' Physica A, 322, pp. 169-79.
The actual spectra of discrete fractional Gaussian noise (dFGN) for frequencies near zero vary as f^{1-2H}, where 0< H < 1 is the Hurst coefficient; however, this form for the spectra need not be a good approximation at other frequencies. When H approaches zero, dFGN spectra exhibit the 1-2H power-law behavior only over a range of low frequencies that is vanishingly small. When dealing with a time series of finite length drawn from a dFGN process with unknown H, practitioners must deal with estimated spectra in lieu of actual spectra. The most basic spectral estimator is the periodogram. The expected value of the periodogram for dFGN with H small also exhibits non-power law behavior. At the lowest Fourier frequencies associated with a time series of N values sampled from a dFGN process, the expected value of the periodogram for H approaching zero varies as f^0 rather than f^{1-2H}. For finite N and small H, the expected value of the periodogram can in fact exhibit a local power law behavior with a spectral exponent of 1-2H at only two distinct frequencies.
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