Probabilistic Stable Functions on Discrete Cones are Power Series (long version)

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