Paretian Poisson processes

Many random populations can be modeled as a countable set of points scattered randomly on the positive half-line. The points may represent magnitudes of earthquakes and tornados, masses of stars, market values of public companies, etc. In this article we explore a specific class of random such populations we coin 'Paretian Poisson processes'. This class is elemental in statistical physics - connecting together, in a deep and fundamental way, diverse issues including: the Poisson distribution of the Law of Small Numbers; Paretian tail statistics; the Frechet distribution of Extreme Value Theory; the one-sided Levy distribution of the Central Limit Theorem; scale-invariance, renormalization and fractality; resilience to random perturbations.