Given a sequence of independent r.v's $\{X_n\}_{n\geq 1}$ such that
$P(X_n=x)=\frac{1}{2}$ if $x=-1$ and/or $x=1$
Let $N\in Po(\lambda)$ be independent of $\{X_n\}_{n\geq 1}$ and we set that $Y=X_1+X_2+\dots+X_N$
One have to show that $\frac{Y}{\sqrt{\lambda}}$ converges in distribution to $N(0,1)$ as $\lambda$ goes to infinity.
How can this be shown? I have been thinking of some kind of use of central limit theorem but I don't get any idea for how to use it to show the convergence...