On the structure of the Wishart distribution

In this paper it is shown that every nonnegative definite symmetric random matrix with independent diagonal elements and at least one nondegenerate nondiagonal element has a noninfinitely divisible distribution. Using this result it is established that every Wishart distribution Wp(k, [Sigma], M) with both p and rank ([Sigma]) >= 2 is noninfinitely divisible. The paper also establishes that any Wishart matrix having distribution Wp(k, [Sigma], 0) has the joint distribution of its elements in the rth row and rth column to be infinitely divisible for every r = 1,2,...,p.