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 (μ_n : n ≥ 0) a sequence of probability measures on G. Say that μ_n admits a Skorokhod representation if, on some probability space, there are random variables X_n with values in (S, G) such that
X_n ~ μ_n for each n ≥ 0 and X_n → X₀ in probability.
We focus on results of the following type: μ_n has a Skorokhod representation if and only if J(μ_n, 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.