The KLR-theorem revisited

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.