A survey on Skorokhod representation theorem without separability

Patrizia Berti, Luca Pratelli, Pietro Rigo
Theory of Stochastic Processes
Vol.20 (36), no.2, 2015, pp.1-12
Let S be a metric space, G a σ-field of subsets of S and (μn : n ≥ 0) a sequence of probability measures on G. Say that μn admits a Skorokhod representation if, on some probability space, there are random variables Xn with values in (S, G) such that
Xn ~ μn for each n ≥ 0 and Xn → X0 in probability.
We focus on results of the following type: μn has a Skorokhod representation if and only if J(μn, μ0) → 0, where J is a suitable distance (or discrepancy index) between probabilities on G. One advantage of such results is that, unlike the usual Skorokhod representation theorem, they apply even if the limit law μ0 is not separable. The index J is taken to be the bounded Lipschitz metric and the Wasserstein distance.