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 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mn>1</mml:mn> <mml:mrow> <mml:mo>/</mml:mo> </mml:mrow> <mml:msup> <mml:mi>f</mml:mi> <mml:mi>β</mml:mi> </mml:msup> </mml:mrow> </mml:math> 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 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mrow> <mml:msub> <mml:mi>P</mml:mi> <mml:mi>τ</mml:mi> </mml:msub> <mml:mo stretchy="false">(</mml:mo> <mml:mi>τ</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mo>∼</mml:mo> <mml:mn>1</mml:mn> <mml:mrow> <mml:mo>/</mml:mo> </mml:mrow> <mml:msup> <mml:mi>τ</mml:mi> <mml:mi>α</mml:mi> </mml:msup> </mml:mrow> </mml:mrow> </mml:math> with abrupt and smooth cutoffs. The asymptotic scaling relation <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mi>β</mml:mi> <mml:mo>=</mml:mo> <mml:mn>3</mml:mn> <mml:mo>−</mml:mo> <mml:mi>α</mml:mi> </mml:mrow> </mml:math> is demonstrated for <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mn>1</mml:mn> <mml:mo>&lt;</mml:mo> <mml:mi>α</mml:mi> <mml:mo>&lt;</mml:mo> <mml:mn>3</mml:mn> </mml:mrow> </mml:math> in the limit of an infinitely broad distribution <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:msub> <mml:mi>P</mml:mi> <mml:mi>τ</mml:mi> </mml:msub> <mml:mo stretchy="false">(</mml:mo> <mml:mi>τ</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> </mml:math> . 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 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mi>β</mml:mi> <mml:mo>=</mml:mo> <mml:mn>3</mml:mn> <mml:mo>−</mml:mo> <mml:mi>α</mml:mi> </mml:mrow> </mml:math> relation are clarified.