Anatomy of a Domain of Continuous Random Variables II

Abstract

In this paper we conclude a two-part analysis of recent work of Jean Goubault-Larrecq and Daniele Varacca, who devised a model of continuous random variables over bounded complete domains. Their presentation leaves out many details, and also misses some motivations for their construction. In this and a related paper we attempt to fill in some of these details, and in the process, we discover a flaw in the model they built.

Our earlier paper showed how to construct $\Theta\textsf{Prob} (A^\infty)$, the bounded complete algebraic domain of thin probability measures over A^∞, the monoid of finite and infinite words over a finite alphabet A. In this second paper, we apply our earlier results to construct $\Theta\textsl{RV}_{A^\infty}(D)$, the bounded complete domain of continuous random variables defined on supports of thin probability measures on A^∞ with values in a bounded complete domain D, and we show $D\mapsto \Theta\textsl{RV}_{A^\infty}(D)$ is the object map of a monad. In the case A = {0,1}, our construction yields the domain of continuous random variables over bounded complete domains devised by Goubault-Larrecq and Varacca. However, we also show that the Kleisli extension $h^\dag\colon {\Theta\textsl{RV}_{A^\infty}}(D)\to {\Theta\textsl{RV}}(E)$ of a Scott-continuous map h : D → E is not Scott continuous, so the construction does not yield a monad on BCD, the category of bounded complete domains and Scott-continuous maps. We leave the question of whether the construction can be rescued as an open problem.