Skorohod representation theorem via disintegrations

Abstract

Let (µ _n : n ≥ 0) be Borel probabilities on a metric space S such that µ _n → µ₀ weakly. Say that Skorohod representation holds if, on some probability space, there are S-valued random variables X _n satisfying X _n ∼ µ _n for all n and X _n → X ₀ in probability. By Skorohod’s theorem, Skorohod representation holds (with X _n → X ₀ 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(µ _n , 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.