Scale-invariance of random populations: From Paretian to Poissonian fractality

Random populations represented by stochastically scattered collections of real-valued points are abundant across many fields of science. Fractality, in the context of random populations, is conventionally associated with a Paretian distribution of the populationʹs values. Using a Poissonian approach to the modeling of random populations, we introduce a definition of “Poissonian fractality” based on the notion of scale-invariance. This definition leads to the characterization of four different classes of Fractal Poissonian Populations—three of which being non-Paretian objects. The Fractal Poissonian Populations characterized turn out to be the unique fixed points of natural renormalizations, and turn out to be intimately related to Extreme Value distributions and to Lévy Stable distributions.