Convergence in distribution (weak convergence) of sum of real-valued random variables

Suppose that $\{(X_n,Y_n)\}^\infty_{n=1}$ is a sequence of pairs of real-valued random variables that converge in distribution to $(X,Y)$. Show that $X_n + Y_n$ converges in distribution to X+Y.

Attempt at a solution: If $h(x,y) = x+y$, then $h(x,y)$ is continuous, and so is any $f(h(x,y))$ where $f$ is a continuous function. I know that the class of bounded, continuous functions is measure-determining, which I think would be useful, but I just can't wrap my head around how to apply it.

Solution 1:

You are very close.

Let $f:\mathbb{R}\rightarrow\mathbb{R}$ an arbitrary continuous and bounded function.

Note that the function $h:\mathbb{R}^{2}\rightarrow\mathbb{R}$ prescribed by $\langle x,y\rangle\mapsto x+y$ is continuous so that the composition $f\circ h:\mathbb{R}^{2}\rightarrow\mathbb{R}$ is continuous.

Also $f\circ h$ is bounded, since $f$ is bounded.

Then $\lim_{n\rightarrow\infty}\mathbb{E}f\left(h\left(X_{n},Y_{n}\right)\right)=\mathbb{E}f\left(h\left(X,Y\right)\right)$ as a consequence of $\left(X_{n},Y_{n}\right)\stackrel{d}{\rightarrow}\left(X,Y\right)$.

This comes to the same as $\lim_{n\rightarrow\infty}\mathbb{E}f\left(X_{n}+Y_{n}\right)=\mathbb{E}f\left(X+Y\right)$.

This is true for any continuous and bounded function $f:\mathbb{R}\rightarrow\mathbb{R}$, so we are allowed to conclude that $X_{n}+Y_{n}\stackrel{d}{\rightarrow}X+Y$.