Central limit theorem - finding cumulants

[LmdL]

Homework Statement

Given:
$$y=\frac{\sum_{i}x_i-N\left \langle x \right \rangle}{\sqrt{N}}$$

Show that the cumulants of y are:
$$\begin{matrix} \left \langle y \right \rangle_c=0& & \left \langle y^2 \right \rangle_c=\left \langle x^2 \right \rangle_c & & \left \langle y^m \right \rangle_c=\left \langle x^m \right \rangle_c N^{1-m/2}\begin{matrix} & for & m>2 \end{matrix} \end{matrix}$$

Homework Equations

Generating function:
$$\tilde{p}\left ( k \right )=\sum_{n=0}^{\infty }\frac{\left ( -ik \right )^n}{n!}\left \langle x^n \right \rangle$$
Cumulant generating function:
$$ln \left ( \tilde{p}\left ( k \right ) \right )=\sum_{n=0}^{\infty }\frac{\left ( -ik \right )^n}{n!}\left \langle x^n \right \rangle_c$$

For independent
$$X=\left ( x_1,x_2,...,x_N \right )$$
if
$$y=a_0+a_1x_1+a_2x_2+...+a_Nx_N$$
then:
$$\tilde{p}_y\left ( k \right )=e^{-ika_0}\prod_{i}\tilde{p}_i\left ( a_ik \right )$$

The Attempt at a Solution

In my case:
$$y=\frac{\sum_{i}x_i-N\left \langle x \right \rangle}{\sqrt{N}}=-\frac{N\left \langle x \right \rangle}{\sqrt{N}}+\frac{1}{\sqrt{N}}x_1+...+\frac{1}{\sqrt{N}}x_N$$
that is:
$$\begin{matrix} a_0=-\frac{N\left \langle x \right \rangle}{\sqrt{N}} & & a_1=...=a_N=\frac{1}{\sqrt{N}} \end{matrix}$$
Substituting into
$$\tilde{p}_y\left ( k \right )=e^{-ika_0}\prod_{i}\tilde{p}_i\left ( a_ik \right )$$
gives:
$$\tilde{p}_y\left ( k \right )=e^{\frac{ikN\left \langle x \right \rangle}{\sqrt{N}}}\prod_{i}\tilde{p}_i\left ( \frac{1}{\sqrt{N}}k \right )$$
Cumulant generating function:
$$ln \left ( \tilde{p}_y\left ( k \right ) \right )=\frac{ikN\left \langle x \right \rangle}{\sqrt{N}}+\sum_{i}ln\left ( \tilde{p}_i\left ( \frac{1}{\sqrt{N}}k \right ) \right )$$
Next step I tried is to represent ln function as series:
$$\sum_{n=0}^{\infty }\frac{\left ( -ik \right )^n}{n!}\left \langle y^n \right \rangle_c = \frac{ikN\left \langle x \right \rangle}{\sqrt{N}}+\sum_{i}^{N}\sum_{n=0}^{\infty }\frac{\left ( -i\frac{1}{\sqrt{N}}k \right )^n}{n!}\left \langle x^n \right \rangle_c=\frac{ikN\left \langle x \right \rangle}{\sqrt{N}}+N\sum_{n=0}^{\infty }\frac{\left ( -i\frac{1}{\sqrt{N}}k \right )^n}{n!}\left \langle x^n \right \rangle_c$$
And, finally, here I'm stuck. Can someone help me how to continue from here?
Thanks in advance.

 

[LmdL]

Doesn't matter. Already solved it.