An Information-Theoretic Proof of the Kac--Bernstein Theorem

TL;DR

This paper provides an information-theoretic proof of the Kac--Bernstein theorem, demonstrating that independence of sums and differences of independent variables implies normality.

Contribution

It introduces a novel proof technique based on information theory for a classical characterization of normal distributions.

Findings

Independence of X+Y and X−Y implies X and Y are normal.

The proof offers new insights into distribution characterization.

Connects information theory with classical probability results.

Abstract

A short, information-theoretic proof of the Kac--Bernstein theorem, which is stated as follows, is presented: For any independent random variables $X$ and $Y$, if $X+Y$ and $X-Y$ are independent, then $X$ and $Y$ are normally distributed.