@article{ECP2793,
author = {Patrizia Berti and Luca Pratelli and Pietro Rigo},
title = {A Skorohod representation theorem without separability},
journal = {Electron. Commun. Probab.},
fjournal = {Electronic Communications in Probability},
volume = {18},
year = {2013},
keywords = {Convergence of probability measures, Perfect probability measure, Separable probability measure, Skorohod representation theorem, Uniform distance},
abstract = {Let $(S,d)$ be a metric space, $\mathcal{G}$ a $\sigma$-field on $S$ and $(\mu_n:n\geq 0)$ a sequence of probabilities on $\mathcal{G}$. Suppose $\mathcal{G}$ countably generated, the map $(x,y)\mapsto d(x,y)$ measurable with respect to $\mathcal{G}\otimes\mathcal{G}$, and $\mu_n$ perfect for $n>0$. Say that $(\mu_n)$ has a Skorohod representation if, on some probability space, there are random variables $X_n$ such that
\begin{equation*}
X_n\sim\mu_n\text{ for all }n\geq 0\quad\text{and}\quad d(X_n,X_0)\overset{P}\longrightarrow 0.
\end{equation*}
It is shown that $(\mu_n)$ has a Skorohod representation if and only if
\begin{equation*}
\lim_n\,\sup_f\,\left|\mu_n(f)-\mu_0(f)\right|=0,
\end{equation*}
where $\sup$ is over those $f:S\rightarrow [-1,1]$ which are $\mathcal{G}$-universally measurable and satisfy $\left|f(x)-f(y)\right|\leq 1\wedge d(x,y)$. An useful consequence is that Skorohod representations are preserved under mixtures. The result applies even if $\mu_0$ fails to be $d$-separable. Some possible applications are given as well.},
pages = {no. 80, 1-12},
issn = {1083-589X},
doi = {10.1214/ECP.v18-2793},
url = {http://ecp.ejpecp.org/article/view/2793}}