# convolution of random variables

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?

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You specification is incomplete. Do you mean that these random variables have a common density? I assume it's supported on $[0,\infty)$. You may only use convolution if the random variables are independent. – ncmathsadist Jul 8 '11 at 20:10
You haven't attempted to specify the JOINT distribution. For example, are they independent? The meaning of your notation is something we have to guess at. My guess is that you intended $2\Omega x e^{-\Omega x^2}$ on the interval $(0,\infty)$ to be the density. – Michael Hardy Jul 8 '11 at 20:52
BTW: 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") – leonbloy Jul 8 '11 at 20:55

## 1 Answer

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.

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