# Conditions for two limiting distributions to be independent

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?

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.
• My bad, I was just wondering how can we construct $(X´,Y´)$. I guess i will think about it – user128422 Dec 7 '19 at 20:32