Fluctuation Scaling and 1/f Noise
Journal of Basic and Applied Physics
A power law relationship between the variance and the mean, when derived from sequential data using expanding enumerative bins, implies 1/f noise. This relationship, called fluctuation scaling by physicists and Taylor's law by ecologists, is found within diverse physical, econometric and biological systems. Its origin remains controversial. Both fluctuation scaling and 1/f noise are proposed to manifest consequent to a central limit-like effect specified by the Tweedie convergence theorem that
... gence theorem that has as its foci of convergence a family of statistical distributions, the Tweedie exponential dispersion models. An example of fluctuation scaling and 1/f noise is provided here based on deviations in position of the prime numbers; the Tweedie compound Poisson distribution is shown to correspond to these deviations. Whereas many different physical and biological mechanisms have been proposed to explain fluctuation scaling, Taylor's law and 1/f noise, such mechanisms are inapplicable to a number theoretic example like this. The Tweedie convergence theorem provides a generally applicable explanation for the origin of these scaling relationships, and can provide insight into processes like self-organized criticality and multifractality.