Some properties and characterizations for generalized multivariate Pareto distributions

In this paper, several distributional properties and characterization theorems of the generalized multivariate Pareto distributions are studied. It is found that the multivariate Pareto distributions have many mixture properties. They are mixed either by geometric, Weibull, or exponential variables. The multivariate Pareto, MP(k)(I), MP(k)(II), and MP(k)(IV) families have closure property under finite sample minima. The MP(k)(III) family is closed under both geometric minima and geometric maxima. Through the geometric minima procedure, one characterization theorem for MP(k)(III) distribution is developed. Moreover, the MP(k)(III) distribution is proved as the limit multivariate distribution under repeated geometric minimization. Also, a characterization theorem for the homogeneous MP(k)(IV) distribution via the weighted minima among the ordered coordinates is developed. Finally, the MP(k)(II) family is shown to have the truncation invariant property.