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

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.