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Thread: characteristic function of linnik distrib

  1. #1
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    characteristic function of linnik distrib

    Let $\displaystyle X_{i} \epsilon Linnik(\alpha), N \epsilon Fs(p), S_n=\sum_{i=1}^n X_i$
    with $\displaystyle \phi_{linnik(\alpha)}= (1+|t|^{\alpha})^{-1}$
    Now show that $\displaystyle p^{(1/\alpha)}S_n$ is again Linnik distributed.

    So i need to show that $\displaystyle p^{(1/\alpha)}S_n$ has characteristic funtion of the form$\displaystyle \phi_{linnik(\alpha)}= (1+|t|^{\alpha})^{-1}$

    I know that $\displaystyle \phi_{S_n}=g_N(\phi_{X(t)})$ with$\displaystyle g_N$ the generating function of N

    So i plug in nad get $\displaystyle g_N(t)=E(t^N)=\sum_n t^n*P(N=n)=\sum_n t^n*p(1-p)^{n-1}$
    $\displaystyle =\frac{p}{(1-p)}\sum_n (t(1-p))^n=\frac{p}{(1-p)}\frac{1}{(1-(1-p)t)}$
    But even this is questionably since i dont know if t(1-p)<1. and even if the former is correct it does not yield the correct answer so where did i go wrong?
    (i get $\displaystyle \phi_{p^{1/\alpha}*S_n}=p (1/(1 - p) + 1/(p + Abs[t]^a))$)
    Last edited by isarutas; Nov 1st 2009 at 07:48 AM.
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