On strong unimodality of multivariate discrete distributions

A discrete function ff defined on ZnZn is said to be logconcave if f(λx+(1−λ)y)≥[f(x)]λ[f(y)]1−λ for x, y, λx+(1−λ)y∈Zn. A more restrictive notion is strong unimodality. Following Barndorff-Nielsen [O. Barndorff-Nielsen, Unimodality and exponential families, Commun. Statist. 1 (1973) 189–216] a discrete function p(z),z∈Zn is called strongly unimodal if there exists a convex function f(x),x∈Rn such that f(x)=−logp(x)  if x∈Zn. In this paper sufficient conditions that ensure the strong unimodality of a multivariate discrete distribution, are given. Examples of strongly unimodal multivariate discrete distributions are presented.