Convergence in distribution of sum of independent random variable, with no independent limit

Question

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).

Answer 1

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].

Answer 2

Well, you can see that the characteristic functions of $X_n+Y_n$ converge pointwise to $\varphi_X\varphi_Y=\varphi_{X’+Y’}$, where $X’$ and $X$ have the same law, and $Y’$ and $Y$ have tre same law, and $X’$ and $Y’$ are independent. From Levi’s theorem, $X_n+Y_n \rightarrow X’+Y’$ in distribution, so $X_n+Y_n \rightarrow X+Y$ in distribution iff $X’+Y’$ and $X+Y$ have the same law.

In a nutshell, $X_n+Y_n$ converges to the “independent sum” of $X$ and $Y$.