A Note on Matrix Variate Normal Distribution

Let X1 and X2 be two identically distributed random variables. Suppose that X1 | X2=x2 has a N(ax2+b, _) distribution for all x2 # R, where & 0. Ahsanullah (1985) showed that these conditions imply |a|<1, X1 , and X2 have a common normal distribution with mean b (1&a;) and variance _ (1&a;), and the joint distribution of X1 and X2 is a bivariate normal distribution with covariance a_ (1&a;). Castillo and Galambos (1989) presented a unified extension of the characterizations of the bivariate normal distribution given by Brucker (1979) and Ahsanulla (1985). In his paper, Ahsanullah also proposed a conjecture on a multidimensional version of his result. Hamedani (1988), then Arnold and Pourahmadi (1988), gave counterexamples to this conjecture, and they also gave different characterizations for multivariate normal distribution based on conditional normality. For the details of these results, see the survey paper of Hamedani (1992). In this note we give other multidimensional versions of the result of Ahsanullah (1985), then apply these results to the characterization of a matrix variate normal distribution with identically distributed row vectors. Our technique is similar to the technique used by Ahsanullah in the bivariate case, combining it with some well known results on matrices and linear transformations on a real Euclidean space R. The basic results on matrices and linear transformations used in the proof of Theorem 2.1 can be found in Halmos (1974) or Young and article no. MV961649