Some characterizations of the multivariate t distribution

A multivariate t vector X is represented in two different forms, one associated with a normal vector and an independent chi-squared variable, and the other with a normal vector and an independent Wishart matrix. We show that X is multivariate t with mean [mu], covariance matrix [nu]([nu] - 2)-1[Sigma], [nu] > 2 and degrees of freedom [nu] if and only if for any a [not equal to] 0, (a'[Sigma]a)-1/2a'(X - [mu]) has the Student's t distribution with [nu] degrees of freedom under both representations. Some other characterizations are also obtained.