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.