TITLE: sum of two random variable converges in distribution
QUESTION [5 upvotes]: It's simple question.
$\{X_n\}, X, Y$ are random variables such that $X_n + cY$ converges to $X+cY$ in distribution for each $c >0$.
How can I show that $X_n$ converges to $X$ in distribution??
I know that $X_n$ converges in distribution to $X$ if and only if $P(X_n \leq x) \rightarrow P(X \leq x)$ at every $x$ where $F_X$ is continuous. But cdfs do not seem to help with this problem. I also tried showing that $E[f(X_n)] \to E[f(X)]$ where $f$ is bounded and continuous, but this leads to double limits ($n\to \infty$ and $c\to 0$).

REPLY [3 votes]: Here's a solution via characteristic functions.
For $(t_1,t_2)\in \mathbb R^2$, assuming $t_1\neq 0$,
$$\begin{align}
\phi_{(X_n,Y)}(t_1,t_2) &= E\Big[\exp\big(i(t_1X_n + t_2Y)\big)\Big] \\
&= E\Big[\exp\big(it_1(X_n + \frac{t_2}{t_1}Y)\big)\Big] \\
&= \phi_{X_n + (t_2/t_1) Y}(t_1)\\
&\xrightarrow[n\to \infty ]{} \phi_{X + (t_2/t_1) Y}(t_1)
= \phi_{(X,Y)}(t_1,t_2)
\end{align}$$
and the convergence still holds true when $t_1=0$.
By Levy's theorem, $(X_n,Y)$ converges in distribution to $(X,Y)$ and by the continuous mapping theorem, $X_n\xrightarrow{d}X$.