Distribution of the product of a Wishart matrix and a normal vector

We consider the distribution of the product of a Wishart matrix and a normal vector with uncommon covariance matrices. We derive the stochastic representation which reduces the computational burden for the generation of realizations of the product. Using this representation, the density function and higher order moments of the product are derived. In a numerical illustration, we investigate some properties of the distribution of the product. We further suggest the Edgeworth type expansions for the product, and we observe that the suggested approximations provide a good performance for moderately large degrees of freedom of a Wishart matrix.