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/<i>f^β</i> for frequency f and a range of exponents <i>β</i> 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 <i>P_τ (τ)</i> ∼ 1/<i>τ^α</i> with abrupt and smooth cutoffs. The asymptotic scaling relation <i>β</i> = 3−α is demonstrated for 1 < α < 3 in the limit of an infinitely broad distribution <i>P_τ (τ)</i>. Logarithmic corrections to the frequency scaling are exposed at the boundaries of the long-range dependence regime, <i>β</i> =0 and <i>β</i> =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 <i>β</i> = 3−α relation are clarified.