Long-range correlations with finite-size effects from a superposition of uncorrelated pulses with power-law distributed durations
M. A. Korzeniowska, O. E. ̃Garcia · Journal of Statistical Mechanics Theory and Experiment · 2025
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 ) ∼ 1 / 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.