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

1. Jun 13, 2012

### S_David

Hello,

I need to find the CDF of

$$\mathcal{X}=\sum_{l=0}^L|h(l)|^2$$

where

$$h(l)$$

is complex Gaussian with zeros mean and variance

$$\sigma^2_l$$

In particular, I need to proof that:

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

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

I found the expression which is:

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

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:

$$b^{L+1}$$

How can I do that?

Thanks