Probabilistic Stable Functions on Discrete Cones are Power Series

We study the category Cstabm of measurable cones and measurable stable functions—a denotational model of an higher-order language with continuous probabilities and full recursion [8]. We look at Cstabm as a model for discrete probabilities, by showing the existence of a full and faithful functor preserving cartesian closed structure which embeds probabilistic coherence spaces—a fully abstract denotational model of an higher language with full recursion and discrete probabilities [7]—into Cstabm. The proof is based on a generalization of Bernstein’s theorem in real analysis allowing to see stable functions between discrete cones as generalized power series.