On a General Theorem of Number Theory Leading to the Gibbs, Bose--Einstein, and Pareto Distributions as well as to the Zipf--Mandelbrot Law for the Stock Market

TL;DR

This paper introduces a new set density concept and proves a general theorem that unifies and refines key statistical distributions like Gibbs, Bose-Einstein, Pareto, and Zipf law, with applications to stock market analysis.

Contribution

It presents a novel set theory theorem that unifies and refines major statistical distributions relevant to various fields, including finance.

Findings

Unified framework for Gibbs, Bose-Einstein, Pareto, and Zipf distributions

Refined mathematical understanding of these distributions

Potential applications to stock market modeling

Abstract

The notion of density of a finite set is introduced. We prove a general theorem of set theory which refines the Gibbs, Bose--Einstein, and Pareto distributions as well as the Zipf law.