On Uniform and α-Monotone Discrete Distributions

Abstract

In this partly expository article, I am concerned with some simple yet fundamental aspects of discrete distributions that are either uniform or have α-monotone probability mass functions. In the univariate case, building on work of F.W. Steutel published in 1988, I look at Khintchine’s theorem for discrete monotone distributions in terms of mixtures of discrete uniform distributions, along with similar results for discrete α-monotone distributions. In the multivariate case, I develop a new general family of multivariate discrete distributions with uniform marginal distributions associated with copulas and consider families of multivariate discrete distributions with α-monotone marginals associated with these.