On normal characterizations by the distribution of linear forms, assuming finite variance

If X1 and X2 are independent and identically distributed (i. i. d.) with finite variance, then (X1+X2)/[radical sign]2 has the same distribution as X1 if and only if X1 is normal with mean zero (Pólya [9]). The idea of using linear combinations of i. i. d. random variables to characterize the normal has since been extended to the case where [sigma][infinity]i=1aiXi has the same distribution as X1. In particular if at least two of the ai's are non-zero and X1 has finite variance, then Laha and Lukacs [8] showed that X1 is normal. They also [7] established the same result without the assumption of finite variance. The purpose of this note is to present a different and easier proof of the characterization under the assumption of finite variance. The idea of the proof follows closely the approach used by Pólya in [9]. The same technique is also used to give a characterization of the exponential distribution.