I have been working on the following exercise:
Let $\{X_{k}\}_{k\geq 1}$ be i.i.d. random variables and let $\{\xi_{n}\}_{n\geq 1}$ be Poisson random variables with $\xi_{n}\in Po(n)$. Assume further that the $X_{k}$ and the $\xi_{n}$ are independent of each other.
Compute the characteristic function of $Z_{n}$ where
$$Z_{n}:=\sum_{k=1}^{\xi_{n}}X_{k}$$
Then, assuming that $\mathbb{E}X_{1}=0$ and $\mathbb{E}X_{1}^{2}=1$ prove that the sequence $Z/\sqrt{n}$ converges in distribution.
I was able to prove that $\phi_{Z_{n}}(t)=e^{n(\phi_{X_{1}}(t)-1)}$ without too much difficulty. As each $Z_{n}$ ``looks like" the sum of the first $n$ of the $X_{k}$ I would expect that $Z_{n}/\sqrt{n}$ converges in distribution to $N(0,1)$. The first part of the exercise would suggest that the Levy continuity theorem is to be used. Operating with the belief that $\phi_{Z_{n}}(t/\sqrt{n})$ converges to $e^{(-1/2)t^{2}}$ we would have that
$$\phi_{Z_{n}}(t/\sqrt{n})=e^{n(\phi_{X_{1}}\left(\frac{t}{\sqrt{n}}\right)-1)}\rightarrow e^{-\frac{1}{2}t^{2}}$$
However I have been unable to prove this convergence. Any help would be appreciated.