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.

A Skorohod representation theorem without separability

P. Rigo
2013

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.
P. Berti; L. Pratelli; P. Rigo
File in questo prodotto:
Eventuali allegati, non sono esposti

I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.

Utilizza questo identificativo per citare o creare un link a questo documento: http://hdl.handle.net/11585/734016
 Attenzione

Attenzione! I dati visualizzati non sono stati sottoposti a validazione da parte dell'ateneo

Citazioni
  • ???jsp.display-item.citation.pmc??? ND
  • Scopus 4
  • ???jsp.display-item.citation.isi??? 3
social impact