PROBARILITY AND MATHEMATICAL STATISTICS ON THREE CRA&ACIITERIZATH~)NS OF THE NORnaAL DISTRIBUTION

Abstract. Of a11 the characterizations of Ule normal distribution, three landmarks are the theorems of Bernstein and Skitovitch wncern-ing independence of linear forms and the theorem of Geary concerning independence of sample mean and variance. In this note, ideas from several proofs of these theorems are distilled to give unified proofs which depend mostly on simple probability and characteristic function concepts. I. Introduction. In this note we present proofs of three characterization theorems: those of Bernstein and of Skitovitch concerning independence of linear forms and Geary's Theorem which concerns independence of sample mean and variance. Part of the motivation for this work was the realization that these three important theorems seem not to be very well known amongs