Statistical universality and the method of Poissonian randomizations
Holon Institute of Technology, PO Box 305, Holon 58102, Israel
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Revised: 6 December 2012
Published online: 31 January 2013
In this paper we demonstrate the remarkable effectiveness of Poissonian randomizations in the generation of statistical universality. We do so via a highly versatile spatio-statistical model in which points are randomly scattered, according to a Poisson process, across a general metric space. The points have general independent and identically distributed random physical characteristics. A probe is positioned in space, and is affected by the points. The effect of a given point on the probe is a function of the physical characteristic of the point and the distance of the point from the probe. We determine the classes of Poissonian randomizations – i.e., the spatial Poissonian scatterings of the points – that render the effects of the points invariant with respect to the physical characteristics of the points. These Poissonian randomizations have intrinsic power-law structures, yield statistical robustness, and generate universal statistics including Lévy distributions and extreme-value distributions. In effect, our results establish how “fractal” spatial geometries lead to statistical universality.
© EDP Sciences, Springer-Verlag, 2013