Singularities of Taylor’s power law in the analysis of aggregation measures

Abstract

Taylor’s law is a well-known power law (TPL) for analysing the scaling behaviour of many fluctuating physical phenomena in nature. The scaling exponent $b$ of this law forms the basis of the aggregation process to which a precise probability density function corresponds. In some phenomena, TPL behaviour with periodic components of the aggregates has been observed for small partitions, especially for physical processes characterised by values of $b=1$ where fluctuation-related aggregation processes are supported by Poissonian distributions. We intend to show that for values of $b$ very close to unity it is possible to find a trend, in the double logarithmic scale, of the TPL that there are ‘periodic patterns’ (components) between variance and mean. This behaviour is found in other binomial-type distributions, of which the Poissonian is a particular case, with mappings characterised by a variance close to 1.