# A slight generalization of Skorokhod's representation theorem

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?

• I don't understand the question. Either the random variables are given or to be constructed. May 12, 2021 at 18:07
• 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. May 12, 2021 at 20:38