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

Abstract

Long-range correlations manifested as power spectral density scaling $1/f^2$ for frequency $f$ and a range of exponents ${\hat{I}}^2$ are investigated for a superposition of uncorrelated pulses with distributed durations $\ddot{I}„$. 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 $P_{\ddot{I}}„(\ddot{I}„)\sim 1/\ddot{I}{„}^{\hat{I}}\pm$ with abrupt and smooth cutoffs. The asymptotic scaling relation ${\hat{I}}^2=3-\hat{I}\pm$ is demonstrated for $1<\hat{I}\pm <3$ in the limit of an infinitely broad distribution $P_{\ddot{I}}„(\ddot{I}„)$. Logarithmic corrections to the frequency scaling are exposed at the boundaries of the long-range dependence regime, ${\hat{I}}^2=0$ and ${\hat{I}}^2=2$. 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 ${\hat{I}}^2=3-\hat{I}\pm$ relation are clarified.

17 pages, 6 figures, 2 tables. Follow-up of this paper: https://link.aps.org/doi/10.1103/PhysRevResearch.5.L022066 . [v4]: Published in Journal of Statistical Mechanics: Theory and Experiment