Electronic Communications in Probability

A Skorohod representation theorem without separability

Patrizia Berti, Luca Pratelli, and Pietro Rigo

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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<br />\begin{equation*}<br />X_n\sim\mu_n\text{ for all }n\geq 0\quad\text{and}\quad d(X_n,X_0)\overset{P}\longrightarrow 0.<br />\end{equation*}<br />It is shown that $(\mu_n)$ has a Skorohod representation if and only if<br />\begin{equation*}<br />\lim_n\,\sup_f\,\left|\mu_n(f)-\mu_0(f)\right|=0,<br />\end{equation*}<br />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.

Article information

Electron. Commun. Probab., Volume 18 (2013), paper no. 80, 12 pp.

Accepted: 18 October 2013
First available in Project Euclid: 7 June 2016

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Zentralblatt MATH identifier

Primary: 60B10: Convergence of probability measures
Secondary: 60A05: Axioms; other general questions 60A10: Probabilistic measure theory {For ergodic theory, see 28Dxx and 60Fxx}

Convergence of probability measures Perfect probability measure Separable probability measure Skorohod representation theorem Uniform distance

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Berti, Patrizia; Pratelli, Luca; Rigo, Pietro. A Skorohod representation theorem without separability. Electron. Commun. Probab. 18 (2013), paper no. 80, 12 pp. doi:10.1214/ECP.v18-2793. https://projecteuclid.org/euclid.ecp/1465315619

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