A probabilistic walk up power laws

We establish a path leading from Pareto’s law to anomalous diffusion, and present along the way a panoramic overview of power-law statistics. Pareto’s law is shown to universally emerge from “Central Limit Theorems” for rank distributions and exceedances, and is further shown to be a finite-dimensional projection of an infinite-dimensional underlying object — Pareto’s Poisson process. The fundamental importance and centrality of Pareto’s Poisson process is described, and we demonstrate how this process universally generates an array of anomalous diffusion statistics characterized by intrinsic power-law structures: sub-diffusion and super-diffusion, Lévy laws and the “Noah effect”, long-range dependence and the “Joseph effect”, 1 / f noises, and anomalous relaxation.