Scale mixtures of Kotz–Dirichlet distributions

In this paper, we first show that a kk-dimensional Dirichlet random vector has independent components if and only if it is a Kotz Type I Dirichlet random vector. We then consider in detail the class of kk-dimensional scale mixtures of Kotz–Dirichlet random vectors, which is a natural extension of the class of Kotz Type I random vectors. An interesting member of the Kotz–Dirichlet class of multivariate distributions is the family of Pearson–Kotz Dirichlet distributions, for which we present a new distributional property. In an asymptotic framework, we show that the Kotz Type I Dirichlet distributions approximate the conditional distributions of scale mixtures of Kotz–Dirichlet random vectors. Furthermore, we show that the tail indices of regularly varying Dirichlet random vectors can be expressed in terms of the Kotz Type I Dirichlet random vectors.