Archimedes, Gauss, and Stein

TL;DR

This paper explores classical and modern characterizations of the Gaussian distribution, connecting historical insights with Stein's method and the beta-gamma algebra to unify various results in probability theory.

Contribution

It introduces a unified framework linking Archimedes, Maxwell, and Stein's characterizations of Gaussian distributions via the beta-gamma algebra.

Findings

Connections between classical and Stein's characterizations

A general framework involving beta-gamma algebra

Explanation of various Stein's method characterizations

Abstract

We discuss a characterization of the centered Gaussian distribution which can be read from results of Archimedes and Maxwell, and relate it to Charles Stein's well-known characterization of the same distribution. These characterizations fit into a more general framework involving the beta-gamma algebra, which explains some other characterizations appearing in the Stein's method literature.