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

Long-range correlations manifested as power spectral density scaling 1/f(beta) for frequency f and a range of exponents beta are investigated for a superposition of uncorrelated pulses with distributed durations tau. 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-tau(tau) similar to 1/tau(alpha) with abrupt and smooth cutoffs. The asymptotic scaling relation beta = 3-alpha is demonstrated for 1 < alpha < 3 in the limit of an infinitely broad distribution P-tau(tau). Logarithmic corrections to the frequency scaling are exposed at the boundaries of the long-range dependence regime, beta = 0 and beta = 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 beta = 3 - alpha relation are clarified.