Skorohod representation theorem via disintegrations

Abstract

Let (µn: n ≥ 0) be Borel probabilities on a metric space S such that µn → µ0 weakly. Say that Skorohod representation holds if, on some probability space, there are S-valued random variables Xn satisfying Xn ∼ µn for all n and Xn → X0 in probability. By Skorohod’s theorem, Skorohod representation holds (with Xn → X0 almost uniformly) if µ0 is separable. Two results are proved in this paper. First, Skorohod representation may fail if µ0 is not separable (provided, of course, non separable probabilities exist). Second, independently of µ0 separable or not, Skorohod representation holds if W(µn, µ0) → 0 where W is Wasserstein distance (suitably adapted). The converse is essentially true as well. Such a W is a version of Wasserstein distance which can be defined for any metric space S satisfying a mild condition. To prove the quoted results (and to define W), disintegrable probability measures are fundamental.