I need to compute the pdf of the sum of a bunch of random variables $\sum_{i=0}^{k-1} c_i X_i $ where $X_i \sim 2\Omega x e^{-\Omega x^2}$, $\Omega > 0$ is a parameter and $c_i$ are positive constant real values. If $k$ is large enough, the law of large numbers may be used. However, in my case $k$ is small, ranging from $3$ to $8$ or $9$. Is there any known result about the pdf of the sum (even an approximation) that I can use without doing the convolution in the domain of the generating function?
convolution of random variables
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1$B$TW: you dont have a "convolution of random variables", this is a "sum of random variables" (which density, in certain conditions results in a "convolution of the densities") – 2011-07-08
1 Answers
Your variable distribution is Rayleigh. The sum of independent Rayleigh (I assume they are indepent,) do not have a closed form solution.
A bound for the weighted sum is given here, in terms on the non weighted sum: http://sankhya.isical.ac.in/pdfs/60a2/6883fnl.pdf
You should google "sum of Rayleigh" to get material (there are some restricted papers: http://portal.acm.org/ft_gateway.cfm?id=1552090&type=pdf http://ieeexplore.ieee.org/iel5/4234/30221/01388722.pdf?arnumber=1388722 )
Regarding your observation "If $k$ is large enough, the law of large numbers may be used." - I guess you mean the central limit theorem- bear in mind that that depends on the behaviour of $c_i$. For example, if they decrease exponentially, the CLT cannot be applied.
Bear also in mind that the CLT, as an asymptotic expansion, can be corrected for finite N using a Edgeworth series with a few terms. Not very straightforward, though.