How to show that Poisson sum of iid $X_i\sim$ Rademacher r.v.'s converges to $N(0,1)$ I have a random sum of independent random variables, i.e. $S_N=X_1+X_2+\ldots+X_N$, where $$P(X=1)=P(X=-1)=0.5$$ which I think would make $X_i$ belong to a Rademacher distribution. If $N\sim \mathrm{Po}(\lambda)$ and is independent, how can I show that $$\frac{S_N}{\sqrt{\lambda}}\rightarrow N(0,1)$$ as $\lambda$ goes to infinity?
 A: Recall the characteristic function of the standard normal distribution:
$$\int_{-\infty}^\infty e^{izx}\frac1{\sqrt{2\pi}} e^{-\frac12 x^2}\,\mathsf dx = e^{-\frac12 z^2}. $$
Now, the characteristic function of $X_1$ is $$\varphi_X(z) = \frac12\left(e^{iz} +e^{-iz}\right) = \cos z,$$
and the generating function of $N$ is $$g_N(z) = \sum_{n=0}^\infty z^n\frac{\lambda^n}{n!}e^{-\lambda} = e^{\lambda(z-1)}. $$
Hence the characteristic function of $S_N/\sqrt\lambda$ is 
\begin{align}
\varphi_{S_N/\sqrt\lambda}(z) &= g_N(\varphi_X(z/\sqrt\lambda))\tag1\\
&= g_N(\cos(z/\sqrt\lambda))\\
&= \exp\left(\lambda\left(\cos(z/\sqrt\lambda)-1\right)\right)\\
&=\exp\left(\lambda\left(1-\frac12\frac{z^2}\lambda + O\left(\frac{z^4}{\lambda^2}\right)-1 \right) \right)\\
&\stackrel{\lambda\to\infty}\longrightarrow \exp\left(-\frac12 z^2\right).
\end{align}
To justify $(1)$, we have
$$
\varphi_{S_N/\sqrt\lambda}(z) = \mathbb E\left[e^{iz\left(S_N/\sqrt{\lambda}\right)} \right]
= \mathbb E\left[ e^{i\left(z/\sqrt\lambda\right)S_n}\right]
=\varphi_{S_N}\left(z/\sqrt\lambda\right)
$$
and for any $\theta\in\mathbb C$, 
\begin{align}
\varphi_{S_N}(\theta) &= \mathbb E\left[ e^{i\theta S_N}\right]\\
&= \mathbb E\left[ \mathbb E\left[ e^{i\theta S_N} \mid N\right]\right]\\
&= \sum_{n=0}^\infty \mathbb E\left[ e^{i\theta S_N} \mid N=n\right]\mathbb P(N=n)\\
&= \sum_{n=0}^\infty \mathbb E\left[e^{i\theta S_n} \right]\mathbb P(N=n)\\
&= \sum_{n=0}^\infty \prod_{k=1}^n \mathbb E\left[e^{i\theta X_k} \right]\mathbb P(N=n)\\
&= \sum_{n=0}^\infty \mathbb E\left[e^{i\theta X_1} \right]^n\mathbb P(N=n)\\
&= g_N\left(\mathbb E\left[e^{i\theta X_1}\right]\right)\\
&= g_N\left(\varphi_X(\theta)\right).
\end{align}
