A Skorohod representation theorem without separability

Patrizia Berti (University of Modena and Reggio-Emilia)
Luca Pratelli (Accademia Navale di Livorno)
Pietro Rigo (University of Pavia)


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
X_n\sim\mu_n\text{ for all }n\geq 0\quad\text{and}\quad d(X_n,X_0)\overset{P}\longrightarrow 0.
It is shown that $(\mu_n)$ has a Skorohod representation if and only if
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.

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Pages: 1-12

Publication Date: October 18, 2013

DOI: 10.1214/ECP.v18-2793


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