Proof for showing convergence in distribution with characteristic function

Question

I need help with a proof by using the characteristic function. To show : with $$ X_n \rightarrow X $$ in distribution , follows that also $$ aX_n + b \rightarrow aX +b $$ converges in distribution. $a,b \in \mathbb{R} $

okay,.. so because of $ X_n \rightarrow X $ , it applies for the characteristic function, that $ \phi_{X_n} (t) \rightarrow, \phi_X (t) $ right? I also know that $ \phi_{X1+....+X_n} = \phi_{X_1} * \phi_{X_2}*...*\phi_{X_n} $.

I feel like this is a very easy proof...but what else do i need?

I appreciate any help :-)

Answer 1

Let $\phi_{n}(t)=Ee^{itX_n}$ and $\phi_{X}(t)=Ee^{itX}$. Then $$ E\exp(it(aX_{n}+b))=\exp(itb)\phi_{n}(at) $$ and note that $$ \exp(itb)\phi_{n}(at)\stackrel{n\to\infty}{\to} \exp(itb)\phi_{X}(at) $$ for all $t$ since $X_n\to X$ in distribution. But $\exp(itb)\phi_{X}(at)$ is the c.f of $aX+b$.