A survey on Skorokhod representation theorem without separability

Patrizia Berti ; Luca Pratelli ; Pietro Rigo

Teoriâ slučajnyh processov, Tome 20 (2015) no. 2, pp. 1-12

Résumé

Let $S$ be a metric space, $\mathcal{G}$ a $\sigma$-field of subsets of $S$ and $(\mu_n:n\geq 0)$ a sequence of probability measures on $\mathcal{G}$. Say that $(\mu_n)$ admits a Skorokhod representation if, on some probability space, there are random variables $X_n$ with values in $(S,\mathcal{G})$ such that \begin{equation*} X_n\sim\mu_n\text{ for each }n\ge 0\quad\text{and}\quad X_n\rightarrow X_0\text{ in probability}. \end{equation*} We focus on results of the following type: $(\mu_n)$ has a Skorokhod representation if and only if $J(\mu_n,\mu_0)\rightarrow 0$, where $J$ is a suitable distance (or discrepancy index) between probabilities on $\mathcal{G}$. One advantage of such results is that, unlike the usual Skorokhod representation theorem, they apply even if the limit law $\mu_0$ is not separable. The index $J$ is taken to be the bounded Lipschitz metric and the Wasserstein distance.

Détail

Keywords: Convergence of probability measures, perfect probability measure, separable probability measure, Skorokhod representation theorem
Mots-clés : uniform distance.