Eur. Phys. J. B 2, 525-539 (1998)
https://doi.org/10.1007/s100510050276

Stretched exponential distributions in nature and economy: "fat tails" with characteristic scales

¹ 107 rue Louis Blériot, 92100 Boulogne-Billancourt, France
² Laboratoire de Physique de la Matière Condensée (CNRS UMR 6622) , Université de Nice-Sophia Antipolis, BP 71, 06108 Nice Cedex 2, France
³ Institute of Geophysics and Planetary Physics and Department of Earth and Space Sciences, UCLA, Box 951567, Los Angeles, CA 90095-1567, USA

Corresponding author: ^a sornette@naxos.unice.fr

Received: 20 January 1998
Revised: 27 January 1998
Accepted: 6 February 1998
Published online: 15 April 1998

Abstract

To account quantitatively for many reported "natural" fat tail distributions in Nature and Economy, we propose the stretched exponential family as a complement to the often used power law distributions. It has many advantages, among which to be economical with only two adjustable parameters with clear physical interpretation. Furthermore, it derives from a simple and generic mechanism in terms of multiplicative processes. We show that stretched exponentials describe very well the distributions of radio and light emissions from galaxies, of US GOM OCS oilfield reserve sizes, of World, US and French agglomeration sizes, of country population sizes, of daily Forex US-Mark and Franc-Mark price variations, of Vostok (near the south pole) temperature variations over the last 400 000 years, of the Raup-Sepkoski's kill curve and of citations of the most cited physicists in the world. We also discuss its potential for the distribution of earthquake sizes and fault displacements. We suggest physical interpretations of the parameters and provide a short toolkit of the statistical properties of the stretched exponentials. We also provide a comparison with other distributions, such as the shifted linear fractal, the log-normal and the recently introduced parabolic fractal distributions.