Q: A question regarding Skorohod's representation theorem and joint weak convergence of $(X_{n-i},Y_n)$ Let $(X_n,Y_n)$ be random variables with $(X_n,Y_n) \rightarrow (X,Y)$ in distribution for $n \to \infty$. By Skorohod's representation theorem, there exists a probability space with random variables $X_n',Y_n'$, $X',Y'$:$$(X_n,Y_n) \sim (X_n',Y_n'), \qquad (X,Y) \sim (X',Y'), \qquad (X_n',Y_n') \to (X',Y') \quad \text{a.s.}$$
Fix $i \in \mathbb{N}$. What can we say about the distribution of $(X_{n-i},Y_n)$? E.g. can we get random variables from Skorohod's theorem, such that also:
$$(X_{n-i},Y_n) \sim (X_{n-i}',Y_n')$$
Q: Is there a counterexample, where $(X_n,Y_n) \rightarrow (X,Y)$ in distribution, but $(X_{n-i},Y_n)$ fails to converge in distribution (to some limit)?
Comment: If $(X_n,Y_n)$ are jointly Markov, then I think we get that all $(X_{n-i},Y_n)$ converge whenever $(X_n,Y_n)$ does.
