Statistics as a dynamical attractor

TL;DR

This paper presents a novel dynamical systems approach to representing any statistical distribution as an attractor of coupled ODEs and Liouville equations, offering insights into the micro-mechanisms behind statistical laws.

Contribution

It introduces a non-Newtonian dynamical framework for statistics, linking statistical distributions to attractors of ODE systems and providing a dynamical interpretation of power law behaviors.

Findings

Any statistical distribution can be modeled as an attractor of coupled ODEs and Liouville equations.

The approach offers a micro-mechanistic understanding of how random events evolve into statistical laws.

Power law distributions are supported by specific dynamical behaviors indicating a 'violent reputation'.

Abstract

It is demonstrated that any statistics can be represented by an attractor of the solution to a corresponding systen of ODE coupled with its Liouville equation. Such a non-Newtonian representation allows one to reduce foundations of statistics to better established foundations of ODE. In addition to that, evolution to the attractor reveals possible micro-mechanisms driving random events to the final distribution of the corresponding statistical law. Special attention is concentrated upon the power law and its dynamical interpretation: it is demonstrated that the underlying dynamics supports a " violent reputation" of the power law statistics.