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arXiv · 2410.07249

Long-range correlations with finite-size effects from a superposition of uncorrelated pulses with power-law distributed durations

M. A. Korzeniowska·O. E. Garcia
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Abstract

Long-range correlations manifested as power spectral density scaling $1/f^β$ for frequency $f$ and a range of exponents $β$ are investigated for a superposition of uncorrelated pulses with distributed durations $τ$. Closed-form expressions for the frequency power spectral density are derived for a one-sided exponential pulse function and several variants of bounded and unbounded power-law distributions of pulse durations ${P_τ(τ)\sim1/τ^α}$ with abrupt and smooth cutoffs. The asymptotic scaling relation $β=3-α$ is demonstrated for $1<α<3$ in the limit of an infinitely broad distribution $P_τ(τ)$. Logarithmic corrections to the frequency scaling are exposed at the boundaries of the long-range dependence regime, $β=0$ and $β=2$. Analytically demonstrated finite-size effects associated with distribution truncations are shown to reduce the frequency ranges of scale invariance by several decades. The regimes of validity of the $β=3-α$ relation are clarified.

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M. A. Korzeniowska, O. E. Garcia. 2025-02-28. Long-range correlations with finite-size effects from a superposition of uncorrelated pulses with power-law distributed durations. https://doi.org/10.1088/1742-5468%2Fadb019

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