Good Morning. Let $(X_n)_n$ and $(Y_n)_n$ be sequences of random variables converging in distribution respectively to $X$ and $Y$. Suppose $X_n,Y_n$ are equally distributed but dependent for all $n$, while $X,Y$ are equally distributed and independent. Is it true that $\operatorname{Cov}(X_n,Y_n)\rightarrow 0$ as $n\rightarrow\infty$?
In my specific problem I also have that $X_n$, $Y_n$ and $X_nY_n$ are uniformly integrable. The thing is that I don't know if the joint variables $(X_n,Y_n)$ converges to $(X,Y)$ in distribution.
Thank you for the help.