Infinitely divisible multivariate and matrix Gamma distributions

Classes of multivariate and cone valued infinitely divisible Gamma distributions are introduced. Particular emphasis is put on the cone-valued case, due to the relevance of infinitely divisible distributions on the positive semi-definite matrices in applications. The cone-valued class of generalised Gamma convolutions is studied. In particular, a characterisation in terms of an Itô–Wiener integral with respect to an infinitely divisible random measure associated to the jumps of a Lévy process is established. example of an infinitely divisible positive definite Gamma random matrix is introduced. It has properties which make it appealing for modelling under an infinite divisibility framework. An interesting relation of the moments of the Lévy measure and the Wishart distribution is highlighted which we suppose to be important when considering the limiting distribution of the eigenvalues.