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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2024
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2410.07249 |
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Table of Contents:
- 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.