Let $(Omega,mathcalA,P)$ be a probability space, S a metric space, $mu$ a probability measure on the Borel $sigma$-field of S, and $X_n:Omega ightarrow S$ an arbitrary map, n = 1, 2, . . .. If $mu$ is tight and $X_n$ converges in distribution to $mu$ (in Hoffmann-Jo rgensen's sense), then $Xsimmu$ for some $S$-valued random variable $X$ on $(Omega,mathcalA,P)$. If, in addition, the $X_n$ are measurable and tight, there are $S$-valued random variables $Z_n$ and $X$, defined on $(Omega,mathcalA,P)$, such that $Z_nsim X_n$, $Xsimmu$, and $Z_n_k ightarrow X$ a.s. for some subsequence $(n_k)$. Further, $Z_n ightarrow X$ a.s. (without need of taking subsequences) if $mu\x=0$ for all $x$, or if $P(X_n=x)=0$ for some $n$ and all $x$. When $P$ is perfect, the tightness assumption can be weakened into separability up to extending $P$ to $sigma(mathcalAcupH)$ for some $HsubsetOmega$ with $P^*(H)=1$. As a consequence, in applying Skorohod representation theorem with separable probability measures, the Skorohod space can be taken $((0, 1), sigma(mathcalUcupH),m_H)$, for some $Hsubset (0,1)$ with outer Lebesgue measure 1, where $mathcalU$ is the Borel $sigma$-field on $(0, 1)$ and $m_H$ the only extension of Lebesgue measure such that $m_H(H)=1$.

Skorohod representation on a given probability space

Rigo P.
2007

Abstract

Let $(Omega,mathcalA,P)$ be a probability space, S a metric space, $mu$ a probability measure on the Borel $sigma$-field of S, and $X_n:Omega ightarrow S$ an arbitrary map, n = 1, 2, . . .. If $mu$ is tight and $X_n$ converges in distribution to $mu$ (in Hoffmann-Jo rgensen's sense), then $Xsimmu$ for some $S$-valued random variable $X$ on $(Omega,mathcalA,P)$. If, in addition, the $X_n$ are measurable and tight, there are $S$-valued random variables $Z_n$ and $X$, defined on $(Omega,mathcalA,P)$, such that $Z_nsim X_n$, $Xsimmu$, and $Z_n_k ightarrow X$ a.s. for some subsequence $(n_k)$. Further, $Z_n ightarrow X$ a.s. (without need of taking subsequences) if $mu\x=0$ for all $x$, or if $P(X_n=x)=0$ for some $n$ and all $x$. When $P$ is perfect, the tightness assumption can be weakened into separability up to extending $P$ to $sigma(mathcalAcupH)$ for some $HsubsetOmega$ with $P^*(H)=1$. As a consequence, in applying Skorohod representation theorem with separable probability measures, the Skorohod space can be taken $((0, 1), sigma(mathcalUcupH),m_H)$, for some $Hsubset (0,1)$ with outer Lebesgue measure 1, where $mathcalU$ is the Borel $sigma$-field on $(0, 1)$ and $m_H$ the only extension of Lebesgue measure such that $m_H(H)=1$.
2007
Berti P.; Pratelli L.; Rigo P.
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11585/734946
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