Gauss and the identity function -- a tale of characterizations of the normal distribution

TL;DR

This paper reveals that many characterizations of the normal distribution stem from the fact that its log-density derivative equals the negative identity function, offering a unified perspective and extending to general densities.

Contribution

It demonstrates that the core property of the normal distribution's log-density derivative explains many existing characterizations and generalizes to other densities.

Findings

Unifies various normal distribution characterizations

Highlights the role of the log-density derivative in distribution properties

Provides a method to extend characterizations to general densities

Abstract

The normal distribution is well-known for several results that it is the only to fulfil. The aim of the present paper is to show that many of these characterizations actually follow from the fact that the derivative of the log-density of the normal distribution is the (negative) identity function. This \emph{a priori} very simple yet surprising observation allows a deeper understanding of existing characterizations and paves the way to an immediate extension to a general density $x\mapsto p(x)$ by replacing $-x$ in these results with $(\log p(x))'$.