# A Proof of the Herschel-Maxwell Theorem Using the Strong Law of Large Numbers

## TL;DR

This paper provides a proof of the Herschel-Maxwell theorem using the strong law of large numbers, offering shorter proofs under certain conditions and connecting to Maxwell's characterization via the central limit theorem.

## Contribution

It introduces a novel proof of the Herschel-Maxwell theorem leveraging the strong law of large numbers and explores alternative proofs using the central limit theorem.

## Key findings

- Normal distribution characterized by spherical symmetry and independence
- Shorter proofs under moment assumptions
- Connection to Maxwell's characterization via CLT

## Abstract

In this article, we use the strong law of large numbers to give a proof of the Herschel-Maxwell theorem, which characterizes the normal distribution as the distribution of the components of a spherically symmetric random vector, provided they are independent. We present shorter proofs under additional moment assumptions, and include a remark, which leads to another strikingly short proof of Maxwell's characterization using the central limit theorem.