Limit distribution of $X_n+Y_n$ if $X_n \overset{d}{\longrightarrow}X$, $Y_n \overset{d}{\longrightarrow}Y$ if $(X,Y) \sim N(\mu,\Sigma)$?

Question

Let $X_n, Y_n$ be sequences of RV with $X_n \overset{d}{\longrightarrow} X$ and $Y_n \overset{d}{\longrightarrow} Y$ so that $\begin{pmatrix} X\\ Y \end{pmatrix} \sim N\left(\begin{pmatrix} \mu_1\\ \mu_2 \end{pmatrix}, \Sigma\right)$ with $\mu=(\mu_1,\mu_2) \in \mathbb{R}^2, \Sigma \in \mathbb{R}^{2 \times 2}.$

I'd like to show that in this case $X_n+Y_n \overset{d}{\longrightarrow} X+Y$.

I thought this usually only held if $X$ and $Y$ were independent. I haven't been able to find a reliable source for this yet, so if anybody knows of one that would be very helpful, too.

Answer 1

This cannot be true, as the example $X_n=Y_n=X=-Y$ standard normal shows.

If $X$ and $Y$ must be independent, try $X_n=Y_n=X$ with $X$and $Y$ i.i.d. standard normal.

If $X_n$ and $Y_n$ must be independent, try $X_n=X=Y$ and $Y_n=Z$ with $X$ and $Z$ i.i.d. standard normal.