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The CDF of sum of i.ni.d exponential RVs

  1. Jun 13, 2012 #1
    Hello,

    I need to find the CDF of

    [tex]\mathcal{X}=\sum_{l=0}^L|h(l)|^2[/tex]

    where

    [tex]h(l)[/tex]

    is complex Gaussian with zeros mean and variance

    [tex]\sigma^2_l[/tex]

    In particular, I need to proof that:

    [tex]\text{Pr}\left[\mathcal{X}\leq b\right]\doteq b^{L+1}[/tex]

    where dotted equal means in asymptotic sense as b approaches 0.

    I found the expression which is:

    [tex]1-\sum_{l=0}^L\beta_l\exp\left(-\frac{b}{\sigma_l^2}\right)[/tex]

    where betas are coefficients come from partial expansion, but I don't know how to prove that it is in the asymptotic sense equals to:

    [tex]b^{L+1}[/tex]

    How can I do that?

    Thanks
     
  2. jcsd
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