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A Skorohod representation theorem without separability

Dublin Core		PKP Metadata Items	Metadata for this Document
1.	Title	Title of document	A Skorohod representation theorem without separability
2.	Creator	Author's name, affiliation, country	Patrizia Berti; University of Modena and Reggio-Emilia; Italy
2.	Creator	Author's name, affiliation, country	Luca Pratelli; Accademia Navale di Livorno; Italy
2.	Creator	Author's name, affiliation, country	Pietro Rigo; University of Pavia; Italy
3.	Subject	Discipline(s)
3.	Subject	Keyword(s)	Convergence of probability measures, Perfect probability measure, Separable probability measure, Skorohod representation theorem, Uniform distance
3.	Subject	Subject classification	60B10, 60A05, 60A10
4.	Description	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.
5.	Publisher	Organizing agency, location
6.	Contributor	Sponsor(s)
7.	Date	(YYYY-MM-DD)	2013-10-18
8.	Type	Status & genre	Peer-reviewed Article
8.	Type	Type
9.	Format	File format	PDF
10.	Identifier	Uniform Resource Identifier	http://ecp.ejpecp.org/article/view/2793
10.	Identifier	Digital Object Identifier	10.1214/ECP.v18-2793
11.	Source	Journal/conference title; vol., no. (year)	Electronic Communications in Probability; Vol 18
12.	Language	English=en	en
14.	Coverage	Geo-spatial location, chronological period, research sample (gender, age, etc.)
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