Poisson central limit theorem
= Poisson central limit theorem
{c}
If $N_\lambda\sim\operatorname{Pois}(\lambda)$ and $\lambda\to\infty$, then
$$
\frac{N_\lambda-\lambda}{\sqrt\lambda}
\ \xrightarrow{\mathrm d}\ N(0,1).
$$
Indeed, its <characteristic function> is
$$
\exp\!\left(\lambda\left(e^{it/\sqrt\lambda}-1-\frac{it}{\sqrt\lambda}\right)\right)
\longrightarrow e^{-t^2/2}.
$$