# The KLR-theorem revisited

## TL;DR

This paper revisits the KLR-theorem, establishing conditions under which certain conditional expectations imply Gaussianity of variables, and introduces a new characterization involving Gaussian components.

## Contribution

It provides a new characterization of Gaussian variables based on conditional expectation relations, including the first known case involving Gaussian components.

## Key findings

- For n≥3 and ab>0, the relation holds iff variables are Gaussian.
- A new characterization arises when a=1, b=-1, indicating Gaussian components.
- The work extends understanding of Gaussianity through conditional expectation properties.

## Abstract

For independent random variables $X_1,\ldots, X_n;Y_1,\ldots, Y_n$ with all $X_i$ identically distributed and same for $Y_j$, we study the relation \[E\{a\bar X + b\bar Y|X_1 -\bar X +Y_1 -\bar Y,\ldots,X_n -\bar X +Y_n -\bar Y\}={\rm const}\] with $a, b$ some constants. It is proved that for $n\geq 3$ and $ab>0$ the relation holds iff $X_i$ and $Y_j$ are Gaussian.\\ A new characterization arises in case of $a=1, b= -1$. In this case either $X_i$ or $Y_j$ or both have a Gaussian component. It is the first (at least known to the author) case when presence of a Gaussian component is a characteristic property.