Abstract
Stochastic processes have seen a number of applications in computation, from model checking and verification of probabilistic systems to the emerging area of probabilistic programming. They also arise in Scott’s Stochastic Lambda Calculus and Barker’s Randomized PCF. In this talk, I will "turn the tables" by proving a basic result from stochastic process theory using domain-theoretic techniques. The result in question is Skorohod’s Theorem, which states that every weakly convergent sequence of probability measures on a Polish space can be realized as the images of Lebesgue measure under a sequence of random variables defined on the unit interval that converge almost surely wrt Lebesgue measure to a random variable that realizes the limit measure. Our domain-theoretic generalization shows that each weakly convergent sequence of probability measures on a countably-based bounded complete domain can be realized as the image of Haar measure on the Cantor group that forms the maximal elements of the Cantor tree, via a sequence of Scott-continuous maps from the Cantor tree to the bounded complete domain that converges wrt the Lawson topology to a Scott-continuous map that realizes the limit measure.