Suppose we have two sequences of random variables (not independent) $(X_n) , ( Y_n) $ which jointly converge to a random variable $ (X,Y) $ in distribution. We assume that $X$ and $Y$ are independent. Can we say anything about the convergence of the sum $X_n + Y_n$?
Note 1: Since $X$ and $Y$ are independent we can say that $X_n \Rightarrow X $ and $Y_n\Rightarrow X $.
Note 2: I am aware that, in general, to say that the sum converges to the sum we need independence. The question is, assuming we have asymptotic independence, can we hope to get the same result?