An extension of Kemperman's characterization on k-independence and its application

We present a mixed-type multivariate distributional model generalizing the model of k-independent identically distributed random variables, and give necessary and sufficient conditions for a distribution of a random vector to be the joint distribution of the numbers of successes of q categories in n such dependent trials. The model includes, as particular cases, mixtures of multivariate extremal distributions and models corresponding to ordinal sums of independence and comonotonicity copulas. As an application, we study the extremal properties of order statistics based on random vectors with known multivariate marginals.