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A Skorohod representation theorem without separability. (English) Zbl 1308.60007
Summary: 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 \[ \lim\limits_n\sup\limits_f|\mu_n(f)-\mu_0(f)|=0, \] where \(\sup\) is over those \(f:S\rightarrow [-1,1]\) which are \(\mathcal G\)-universally measurable and satisfy \(|f(x)-f(y)|\leq 1\wedge d(x,y)\). A 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.

60B10 Convergence of probability measures
60A05 Axioms; other general questions in probability
60A10 Probabilistic measure theory
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