The necessary and sufficient conditions for dependent quadratic forms to be distributed as multivariate gamma

AbstractLet S be distributed as noncentral Wishart given by Wp(m, Σ, Ω) and let x be an n × 1 random vector distributed as N(μ, V). If qi = x′Aix + 2l′ix + ci, i = 1, 2,…, p, are p dependent second degree polynomials in the elements of x where Aj's are symmetric matrices, then the necessary and sufficient conditions for q1 , q2 ,…, qp to be distributed as the diagonal elements of S are established and this generalizes the result for Σ = I. Some special cases are considered