In this paper, we first show that a k-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 k-dimensional scale mixtures of Kotz-Dirichlet random vectors, which is a natural extension of the class of Katz 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 Katz Type I Dirichlet random vectors.