Take $X_n$ and $Y_n$ to be two sequence of r.v., and $X,Y$ r.v. such that
1) $X_n \Rightarrow X$ and $Y_n \Rightarrow Y$ in distribution.
2) $X_n$ is independent of $Y_n$ for each $n \in \mathbb{N}$
is that true that $X_n+Y_n\Rightarrow X+Y$.
Note that i'm not requiring $X$ to be independent on $Y$, as assumed in the case treated in this post (Sum of two independent random variables converges in distribution).
Best Answer
Let $(X_n)$ and $(Y_n)$ be i.i.d. standard normal, independent of each other. Then $X_n \to X_1$ in distribution and $Y_n \to -X_1$ in distribution but $X_n+Y_n$ does not tend to $X_1-X_1=0$ in distribution. [ $X_n+Y_n$ has $N(0,2)$ distribution].