, Volume 110, Issue 3-6, pp 739-774

Multifractal Power Law Distributions: Negative and Critical Dimensions and Other “Anomalies,” Explained by a Simple Example

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Abstract

“Divergence of high moments and dimension of the carrier” is the subtitle of Mandelbrot's 1974 seed paper on random multifractals. The key words “divergence” and “dimension” met very different fates. “Dimension” expanded into a multifractal formalism based on an exponent α and a function f(α). An excellent exposition in Halsey et al. 1986 helped this formalism flourish. But it does not allow divergent high moments and the related inequalities f(α)<0 and α<0. As a result, those possibilities did not flourish. Now their time has come for diverse reasons. The broad 1974 definitions of α and f allow α<0 and f(α)<0, but the original presentation demanded to be both developed and simplified. This paper shows that both multifractal anomalies occur in a very simple example, which has been crafted for this purpose. This example predicts the power law distribution. It generalizes α and f(α) beyond their usual roles of being a Hölder exponent and a Hausdorff dimension. The effect is to allow either f or both f and α to be negative, and the apparent anomalies are made into sources of new important information. In addition, this paper substantially clarifies the subtle way in which randomness manifests itself in multifractals.