The KLR-theorem revisited
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For independent random variables X_1,..., X_n;Y_1,..., Y_n with all X_i identically distributed and same for Y_j, we study the relation E{aX̅ + bY̅|X_1 -X̅ +Y_1 -Y̅,...,X_n -X̅ +Y_n -Y̅}= const with a, b some constants. It is proved that for n≥ 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.
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