| Título: | A Skorohod representation theorem without separability |
| Autores: |
Berti, Patrizia; University of Modena and Reggio-Emilia Pratelli, Luca; Accademia Navale di Livorno Rigo, Pietro; University of Pavia |
| Fecha: | 2013-01-03 |
| Publicador: | Electronic communications in probability |
| Fuente: |
 |
| Tipo: | Peer-reviewed Article |
| Tema: |
Convergence of probability measures, Perfect probability measure, Separable probability measure, Skorohod representation theorem, Uniform distance 60B10, 60A05, 60A10 |
| Descripción: | 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. |
| Idioma: | Inglés |
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