### 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}→ X

_{0}in probability.

We focus on results of the following type: μ

_{n}has a Skorokhod representation if and only if J(μ

_{n}, μ

_{0}) → 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 μ

_{0}is not separable. The index J is taken to be the bounded Lipschitz metric and the Wasserstein distance.