Multifractal power law distributions: Negative and critical dimensions and other ”Anomalies”, explained by a simple example

‘‘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 a and a function f(a). 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(a) < 0 and a < 0. As a result, those possibilities did not flourish. Now their time has come for diverse reasons. The broad 1974 definitions of a and f allow a < 0 and f(a) < 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 a and f(a) beyond their usual roles of being a Hölder exponent and a Hausdorff dimension. The effect is to allow either f or both f and a to be negative, and the apparent anomalies are made into sources of new important information. In addition, this paper substantiall