Skorohod Representation Theorem via Disintegrations

摘要

Let (μ_n : n ≥ 0) be Borel probabilities on a metric space S such that μ_n →μ_o 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_o in probability. By Skorohod's theorem, Skorohod representation holds (with X_n → X_o almost uniformly) if μ_o is separable. Two results are proved in this paper. First, Skorohod representation may fail if μ_o is not separable (provided, of course, non separable probabilities exist). Second, independently of μ_o separable or not, Skorohod representation holds if W(μ_n, μ_o) → O 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.