Lacunarity of the zero crossings of Gaussian processes

A lacunarity analysis of the zero-crossings derived from Gaussian stochastic processes with oscillatory autocorrelation functions is evaluated and reveals distinct multi-scaling signatures depending on the smoothness and degree of anti-correlation of the process. These bear qualitative similarities and quantitative distinctions from an oscillatory deterministic signal and a Poisson random process both possessing the same mean interval-size between crossings. At very small and large scales compared with the correlation length of the random processes, the lacunarity is similar to the Poisson but exhibits significant departures from Poisson behaviour if there is a zero-frequency component to the process’s power-spectrum. A comparison of exact results with the gliding box technique that is frequently used to determine lacunarity demonstrates its inherent bias