# Convergence in distribution of Poisson Process

With the central limit theorem, for an i.i.d. process $X_n$ (with mean $m$ and variance $σ^2$), the corresponding normalized sum process is: $$Z_n = \frac{S_n-nm}{σ\sqrt{n}}$$ with $S_n = X_1+X_2+ . . . + X_n$. I know that this does indeed converge in distribution to a zero-mean unit-variance Gaussian. My question is if this is true for the Poisson process and why/why not? I am considering using the taylor expansion of the characteristic function to show whether or not it converges to that of a Gaussian, but I am not quite sure how.

Thanks!

-
Have you computed the characteristic function of $Z_n$ in this case? If so, you should show it in the question. If not, what is stopping you? –  Did Dec 12 '12 at 7:16
@did I have not and was looking for help in doing so –  αδζδζαδζασεβ23τ254634ω5234ησςε Dec 12 '12 at 7:23
You do not say what is stopping you. So... let $N$ be Poisson with parameter $\lambda$, what is the characteristic function of $N$? –  Did Dec 12 '12 at 7:29
@did $e^{λ(e^{it}-1)}$, but I'm not sure how to use that to show convergence –  αδζδζαδζασεβ23τ254634ω5234ησςε Dec 12 '12 at 7:40
Hold on, you will see... Now, what is the characteristic function of $(N-\lambda)/\sqrt{\lambda}$. (Please insert this in your question.) –  Did Dec 12 '12 at 7:42