Five degrees of randomness

Iddo Eliazar

Physica A: Statistical Mechanics and its Applications, 2021, vol. 568, issue C

Abstract: Randomness is omnipresent, and hence the quantification of randomness is a fundamental necessity across the sciences. As “necessity is the mother of invention”, scientists devised various approaches to quantify randomness: statistics uses standard deviation; statistical physics and information theory use entropies (e.g. Shannon); socioeconomics uses inequality indices (e.g. Gini); and ecology uses diversity indices (e.g. Simpson). Alternative to these approaches – which are all continuous quantifications – Mandelbrot suggested a radically different approach: a digital categorization of randomness. Inspired by Mandelbrot, here we showcase a digital categorization comprising five degrees of randomness — á la the Saffir–Simpson hurricane scale, and á la the DEFCON states of defense readiness. Using the reliability-engineering notion of hazard rates, we present a comprehensive study of the digital categorization. From a scholarly viewpoint, we unveil the categorization’s profound connections to Gibbs measures in statistical physics, and to the following probability-theory notions: heavy tails, long tails, slow variation, regular variation, and rapid variation. From an applicative viewpoint, we demonstrate the categorization’s potency and usability. This paper is relevant to wide audiences: theoreticians and practitioners that are tackling random systems and processes.

Keywords: Mild and wild randomness; Hazard rates; Gibbs measures; Heavy and long tails; Slow, regular, and rapid variation (search for similar items in EconPapers)
Date: 2021
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Persistent link: https://EconPapers.repec.org/RePEc:eee:phsmap:v:568:y:2021:i:c:s0378437120309602

DOI: 10.1016/j.physa.2020.125662

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