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Central limit theorem - finding cumulants

  1. Nov 8, 2014 #1
    1. The problem statement, all variables and given/known data
    Given:
    [tex]y=\frac{\sum_{i}x_i-N\left \langle x \right \rangle}{\sqrt{N}}[/tex]

    Show that the cumulants of y are:
    [tex]
    \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}
    [/tex]

    2. Relevant equations
    Generating function:
    [tex]\tilde{p}\left ( k \right )=\sum_{n=0}^{\infty }\frac{\left ( -ik \right )^n}{n!}\left \langle x^n \right \rangle[/tex]
    Cumulant generating function:
    [tex]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[/tex]

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

    3. The attempt at a solution
    In my case:
    [tex]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[/tex]
    that is:
    [tex]\begin{matrix} a_0=-\frac{N\left \langle x \right \rangle}{\sqrt{N}} & & a_1=...=a_N=\frac{1}{\sqrt{N}}
    \end{matrix}[/tex]
    Substituting into
    [tex]\tilde{p}_y\left ( k \right )=e^{-ika_0}\prod_{i}\tilde{p}_i\left ( a_ik \right )[/tex]
    gives:
    [tex]\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 )[/tex]
    Cumulant generating function:
    [tex]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 )[/tex]
    Next step I tried is to represent ln function as series:
    [tex]\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[/tex]
    And, finally, here I'm stuck. Can someone help me how to continue from here?
    Thanks in advance.
     
  2. jcsd
  3. Nov 9, 2014 #2
    Doesn't matter. Already solved it.
     
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