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Let two sequences of random variables $X_n \Rightarrow \mathcal{L}_X$ and $Y_n \Rightarrow \mathcal{L}_Y$, where $\Rightarrow$ denotes convergence in law, can anyone give a general condition on $X_n$, $Y_n$ for $\mathcal{L}_X$ and $\mathcal{L}_Y$ to be independent?

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If $X_n$ and $Y_n$ are independent for all $n\ge 0$, $(X_n,Y_n)\Rightarrow \mathcal{L}_X\otimes \mathcal{L}_Y$. Then one may construct a coupling $(X',Y')$ s.t. $\mathcal{L}_{X'}=\mathcal{L}_X$, $\mathcal{L}_{Y'}=\mathcal{L}_Y$, and $X'$ and $Y'$ are independent.

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  • $\begingroup$ Is there any link or reference for the proof of this result? $\endgroup$ – user128422 Dec 7 '19 at 19:47
  • $\begingroup$ @user128422 What kind of proof are you looking for? It is a simple fact. $\endgroup$ – d.k.o. Dec 7 '19 at 19:48
  • $\begingroup$ My bad, I was just wondering how can we construct $(X´,Y´)$. I guess i will think about it $\endgroup$ – user128422 Dec 7 '19 at 20:32

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