Let \$(S,d)\$ be a metric space, \$mathcalG\$ a \$sigma\$-field on \$S\$ and \$(mu_n:ngeq 0)\$ a sequence of probabilities on \$mathcalG\$. Suppose \$mathcalG\$ countably generated, the map \$(x,y)mapsto d(x,y)\$ measurable with respect to \$mathcalGotimesmathcalG\$, and \$mu_n\$ perfect for \$n&gt;0\$. Say that \$(mu_n)\$ has a Skorohod representation if, on some probability space, there are random variables \$X_n\$ such that eginequation* X_nsimmu_n ext for all ngeq 0quad extandquad d(X_n,X_0)oversetPlongrightarrow 0. endequation* It is shown that \$(mu_n)\$ has a Skorohod representation if and only if eginequation* lim_n,sup_f,absmu_n(f)-mu_0(f)=0, endequation* where \$sup\$ is over those \$f:S ightarrow [-1,1]\$ which are \$mathcalG\$-universally measurable and satisfy \$absf(x)-f(y)leq 1wedge 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.

### A Skorohod representation theorem without separability

#### Abstract

Let \$(S,d)\$ be a metric space, \$mathcalG\$ a \$sigma\$-field on \$S\$ and \$(mu_n:ngeq 0)\$ a sequence of probabilities on \$mathcalG\$. Suppose \$mathcalG\$ countably generated, the map \$(x,y)mapsto d(x,y)\$ measurable with respect to \$mathcalGotimesmathcalG\$, 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 eginequation* X_nsimmu_n ext for all ngeq 0quad extandquad d(X_n,X_0)oversetPlongrightarrow 0. endequation* It is shown that \$(mu_n)\$ has a Skorohod representation if and only if eginequation* lim_n,sup_f,absmu_n(f)-mu_0(f)=0, endequation* where \$sup\$ is over those \$f:S ightarrow [-1,1]\$ which are \$mathcalG\$-universally measurable and satisfy \$absf(x)-f(y)leq 1wedge 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.
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P. Berti; L. Pratelli; P. Rigo
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Utilizza questo identificativo per citare o creare un link a questo documento: `http://hdl.handle.net/11585/734016`
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