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 $1\mathrm{/}{f}^{\beta }$ 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 $P_{\tau }(\tau )\sim 1\mathrm{/}{\tau }^{\alpha }$ 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.