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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.

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