Finite-Sample Effects on the Spectral Characterization and Power- Law Convergence of Fractional Gaussian Noise
Hsing-Cheng Liu
Signals and Telecommunication Journal, Vol. 15, No. 2, 2026.
DOI: https://doi.org/10.6025/stj/2026/15/2/45-63
Link: https://www.dline.info/stj/fulltext/v15n2/stjv15n2_1.pdf
Abstract:
Characterizing long range dependence (LRD) and statistical self similarity in complex systems relies heavily
on accurately estimating the scaling parameters of fractional Gaussian noise (fGn). However, empirical
observations are inherently finite, and short observation windows severely constrain frequency resolution,
introducing spectral leakage and estimation variance that obscure intrinsic fractal scaling. This study
systematically investigates the finite sample effects on the spectral characterization and power law
convergence of fGn. We developed a controlled, scalable benchmark using synthetic fGn realizations across
four dyadic lengths (N = 64 to 65,536). By employing dual spectral estimators Welchâ€™s averaged periodogram
and the raw Fast Fourier Transform we tracked the evolution of the spectral exponent, goodness of fit, and
confidence intervals. Our results demonstrate a non-linear transition from noise dominated, white like spectra
in short records to asymptotic power law scaling in longer sequences. For the shortest realizations, estimation
variance dominates, yielding statistically fragile exponents. Conversely, as sample size increases to N ï‚³
4096, the spectral exponents stabilize near 0.68 (corresponding to a persistent Hurst exponent of ~0.84),
with RÂ² values exceeding 0.75 and significantly narrowed confidence intervals. The concordance between
smoothed and unsmoothed spectral representations confirms the robustness of this convergence. Ultimately,
this work quantifies the minimum observational requirements necessary to reliably distinguish true fractal
scaling from finite data artifacts, providing a rigorous reference framework for the spectral analysis of
longmemory stochastic processes in complex systems.