Cauchy convolution with other distribution

In summary, the conversation discusses the search for a model for a set of data that is likely a convolution of a Cauchy distribution with another distribution. The goal is to find a tractable analytical formula for the other distribution, with defined first and second moments. While the convolution of Cauchy with Cauchy is again Cauchy, the other function in the convolution needs to be determined first. There is a possibility of using the Voigt distribution, which is a convolution of Cauchy with a normal distribution, but there is no analytical formula currently available. The main challenge is finding a non-negative function that satisfies certain integration criteria.
  • #1
DrDu
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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!
 
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  • #2
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
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
DrDu said:
Apparently there is a convolution of Cauchy with a normal distribution called Voigt distribution, but there is no analytical formula available.

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 [itex] g(x) [/itex] such that [itex] \int_{-\infty}^{\infty} g(x) dx [/itex] exists and [itex] \int_{-\infty}^{\infty} \frac{1}{Ax^2 + Bx + C} g(y-x) dx [/itex] has a closed form solution.
 
  • #5
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.
 

1. What is Cauchy convolution and how does it differ from other distributions?

Cauchy convolution is a mathematical operation that combines two probability density functions to create a new probability density function. It differs from other distributions in that it is non-additive and has heavy tails, meaning it assigns a higher probability to extreme events.

2. What is the formula for Cauchy convolution?

The formula for Cauchy convolution is f(x) = ∫ f(t)g(x-t)dt, where f(x) and g(x) are the two probability density functions being convolved.

3. What are some real-world applications of Cauchy convolution?

Cauchy convolution is commonly used in finance and economics to model stock market returns and economic data. It is also used in physics to model particle interactions and in engineering to model extreme events in data.

4. Can Cauchy convolution be applied to any type of distribution?

Yes, Cauchy convolution can be applied to any type of continuous probability distribution, including normal, exponential, and gamma distributions. However, it is most commonly used with distributions that have heavy tails.

5. Are there any limitations or drawbacks to using Cauchy convolution?

One limitation of Cauchy convolution is that it cannot be applied to discrete distributions. Additionally, the resulting Cauchy convolution function may not have a closed-form solution, making it difficult to analyze and work with mathematically.

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