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Let $f:\mathbb{R}^p\rightarrow\mathbb{R}^q$ $(p,q\geq 1)$ be a continuous function and $(X_n)_{n\geq 1}$ a sequence of random values on $\mathbb{R}^p$ such that $f(X_n)$ converges in law to a random value $Y$ on $\mathbb{R}^q$.

Question: Does there exist a probability space and random values $X_n',Y'$ on it having respectively the same distribution as $X_n$ and $Y$, and such that $f(X_n')\longrightarrow Y'$ a.s?

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  • $\begingroup$ I don't understand the question. Either the random variables are given or to be constructed. $\endgroup$ May 12, 2021 at 18:07
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    $\begingroup$ In this form your question does not make sense. It should be: Does there exist a probability space and r.v. $X_n'$, $Y'$ on it such that the $X_n'$ have the same distribution as the $X_n$, $Y'$ the same distribution as $Y$ and with $f(X_n') \to Y'$ a.s. $\endgroup$ May 12, 2021 at 20:38

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