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

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 behavior with periodic components of the aggregates has been observed for small partitions, especially for physical processes characterized 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. • Analysis of Taylor Power Law (TPL) behaviour with periodic variance patterns. • Analysis of differences between variance and mean in TPLs for Binomial-type distributions and for small aggregation values. • A new power law that allows differentiation periodic patterns in TPLs from non- periodic ones.