A Generalized Box-Muller algorithm for heavy-tail random variables

Bill Thistleton and I recently completed an addendum to the Generalized Box-Muller algorithm for generating q-Gaussian. The algorithm provides a simple methodology for generating heavy-tail random variables, which is valuable in modeling complex systems. In 2006 when the algorithm was originally designed and published with Constantino Tsallis and John Marsh, we discovered an interesting transformation between the input specification of the tail decay via the parameter q and the output tail decay of the random variable. While part of the specification for the algorithm, the reason for the transformation was not understood.

In 2015, I published an explanation that the Tsallis entropy parameter q is a composite of three properties and that the tail shape parameter can be isolated as the inverse of the degree of freedom. Today, I refer to this parameter as the "Nonlinear Statistical Coupling" as it measures the magnitude of the nonlinear source generating the heavy-tail phenomena. In our addendum, Bill and I show that the generalized Box-Muller can now be specified with the precise shape of the tail. Furthermore, as these figures show, the tail shape or coupling can be shown diagrammatically as the magnitude of the fluctuation of the scale of the distribution.

Insights in science often take time to develop. Tsallis and the many investigators around the world advancing our knowledge of nonextensive statistical mechanics have contributed invaluable insights regarding the properties of complex systems. I'm hopeful that clarifying the composite properties of the q parameter will both simplify and further advance our ability to model, control, and utilize complex systems. Bill and I would be curious to hear of your own applications impacted by outliers, fluctuating noise, or other processes that are challenging to model and control.

Theoretical and Generalized Box-Muller generated probability densities for coupled Gaussian distributions in the a) heavy-tail and b) compact-support domain.

Link to arXiv paper: