A characterization of the matrix variate normal distribution having identically distributed row vectors based on conditional normality is given.
1997 Academic Press
1. INTRODUCTION AND BASIC RESULTS
Let X 1 and X 2 be two identically distributed random variables. Suppose that X 1 | X 2 =x 2 has a N(ax 2 +b, \_ 2 ) distribution for all x 2 # R, where & 0. Ahsanullah (1985) showed that these conditions imply |a| <1, X 1 , and X 2 have a common normal distribution with mean bร(1&a) and variance \_ 2 ร(1&a 2 ), and the joint distribution of X 1 and X 2 is a bivariate normal distribution with covariance a\_ 2 ร(1&a 2 ). 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 n .
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 148 0047-259Xร97 25.00