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Statistical universality and the method of Poissonian randomizations

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Abstract

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.

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  1. Holon Institute of Technology, PO Box 305, Holon, 58102, Israel

    Iddo Eliazar

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Eliazar, I. Statistical universality and the method of Poissonian randomizations. Eur. Phys. J. Spec. Top. 216, 3–20 (2013). https://doi.org/10.1140/epjst/e2013-01724-4

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