Showing convergence by using Levy's continuity theorem

Question

Let $X_n \sim \mathrm{Po} (n)$. I want to show that $Y_n = \displaystyle \frac{X_n -n}{ \sqrt{n}}$ for $n \rightarrow \infty$ converges in distribution to a random variable that is standard normal distributed by usig Levy's continuity theorem.

So i have to calculate $ \lim_{n \to \infty} \Phi_{Y_n} = e^{ \displaystyle \frac{-t^2}{2}}$, where $\Phi_{Y_n}$ is the characteristic function of $Y_n$. I know that the characteristic function of $X_n$ is given by $e^{eit - 1}$, but I'm struggeling with the one for $Y_n$. Can anybody help?

Thanks in advance!

Answer 1

Hint: using the definition of characteristic function, show that if $\phi(t)$ is the characteristic function of a random variable $X$, then the characteristic function of $(X-a)/b$ is $e^{-ita/b} \phi(t/b)$.

Also, you have the wrong characteristic function for $X_n$; it's not $e^{eit-1}$ but $e^{n(e^{it}-1)}$.