Hello, I am stuck with the following problem:

Given N=n, the conditional distribution of Y is $\displaystyle \chi^2_{2n}.$
The unconditional distribution of N is Poisson($\displaystyle \theta$).
Show that as $\displaystyle \theta \rightarrow \infty, (Y-EY)/\sqrt{Var(Y)} \rightarrow N(0,1)$ in distribution.

Here's what I done so far, I know that is not much...

$\displaystyle EY=E[E(Y|N)]=E[2N]=2EN=2 \theta$


$\displaystyle Var(Y)=E[Var(Y|N)]+Var[E(Y|N)]=E[4N]+Var(2N)=4 \theta + 4 \theta = 8 \theta$


$\displaystyle \frac{Y-EY}{\sqrt{Var(Y)}}=\frac{Y-2 \theta}{\sqrt{8 \theta}}$

Any hints? Thanks in advance.