# A Cauchy convolution with other distribution

1. Jul 18, 2016

### DrDu

I have a set of data which are probably convolutions of a Cauchy distribution with some other distribution. I am looking for some model for this other distribution so that a tractable analytic formula results. I know that the convolution Cauchy with Cauchy is again Cauchy, but I want the other function in the convolution to have defined first and second moment. Apparently there is a convolution of Cauchy with a normal distribution called Voigt distribution, but there is no analytical formula available. Any ideas?

Thank you very much!

2. Jul 18, 2016

### micromass

I'm rather curious how you know it's a convolution with a Cauchy distribution.
Do you have any information about this Cauchy distribution? Is it the standard Cauchy distribution? Do you know its median?

3. Jul 18, 2016

### DrDu

The Cauchy function is a theoretical model for concentration as a function of other concentrations and an equilibrium constant, not a statistical distribution. However I want to fit a distribution of the medians of the equilibrium constant.

4. Jul 19, 2016

### Stephen Tashi

For what are we seeking an analytical formula - for the probability density function of the convolution ?

Does that amount to saying:

Find a non-negative function $g(x)$ such that $\int_{-\infty}^{\infty} g(x) dx$ exists and $\int_{-\infty}^{\infty} \frac{1}{Ax^2 + Bx + C} g(y-x) dx$ has a closed form solution.

5. Jul 19, 2016

### DrDu

Yes, this was my problem. I solved it using a Sips distribution as g, which is a function of ln x rather than x, and a partial fraction decomposition.