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Zbl 1020.60024
Raymond, G.M.; Percival, D.B.; Bassingthwaighte, J.B.
The spectra and periodograms of anti-correlated discrete fractional Gaussian noise. (English)
[J] Physica A 322, No.1-4, 169-179 (2003). ISSN 0378-4371

Summary: Discrete fractional Gaussian noise (dFGN) has been proposed as a model for interpreting a wide variety of physiological data. The form of actual spectra of dFGN for frequencies near zero varies 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 small $H$ 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.

MSC 2000:: *60G15 Gaussian processes
62M15 Spectral analysis of processes

Keywords: Hurst coefficient; finite-length time series; spectral exponent

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Zentralblatt MATH Berlin [Germany]