The Embry conditions are a set of positivity conditions that characterize subnormal operators (on Hilbert spaces) whose theory is closely related to the theory of positive definite functions on the additive semigroup ℕ of non-negative integers. Completely hyperexpansive operators are the negative definite counterpart of subnormal operators. We show that completely hyperexpansive operators are characterized by a set of negativity conditions, which are the natural analog of the Embry conditions for subnormality. While the genesis of the Embry conditions can be traced to the Hausdorff Moment Problem, the genesis of the conditions to be established here lies in the Lévy-Khinchin representation as holding in the context of abelian semigroups. We actually establish the desired negativity criteria for completely hyperexpansive operator tuples.