Characterizations of certain multivariate distributions
Y. H. Wang · Mathematical Proceedings of the Cambridge Philosophical Society · 1974
LetX1,X2, …,Xn, ben(n≥ 2) independent observations on a one-dimensional random variableXwith distribution functionF. Let be the sample mean and be the sample variance. In 1925, Fisher (2) showed that if the distribution functionFis normal then andS2are stochastically independent. This property was used to derive the student'st-distribution which has played a very important role in statistics. In 1936, Geary(3) proved that the independence of andS2is a sufficient condition forFto be a normal distribution under the assumption thatFhas moments of all order. Later, Lukacs (14) proved this result assuming only the existence of the second moment ofF.The assumption of the existence of moments ofFwas subsequently dropped in the proofs given by Kawata and Sakamoto (7) and by Zinger (27). Thus the independence of andS2is a characterizing property of the normal distribution.