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Title: Ubiquity of Benford's law and emergence of the reciprocal distribution

In this paper, we apply the Law of Total Probability to the construction of scale-invariant probability distribution functions (pdf's), and require that probability measures be dimensionless and unitless under a continuous change of scales. If the scale-change distribution function is scale invariant then the constructed distribution will also be scale invariant. Repeated application of this construction on an arbitrary set of (normalizable) pdf's results again in scale-invariant distributions. The invariant function of this procedure is given uniquely by the reciprocal distribution, suggesting a kind of universality. Finally, we separately demonstrate that the reciprocal distribution results uniquely from requiring maximum entropy for size-class distributions with uniform bin sizes.
 [1] ;  [1] ; ORCiD logo [2]
  1. Los Alamos National Lab. (LANL), Los Alamos, NM (United States). Theoretical Division
  2. Harvard Univ., Cambridge, MA (United States). Dept. of Earth and Planetary Sciences; Santa Fe Inst. (SFI), Santa Fe, NM (United States)
Publication Date:
OSTI Identifier:
Report Number(s):
Journal ID: ISSN 0375-9601
Grant/Contract Number:
Accepted Manuscript
Journal Name:
Physics Letters. A
Additional Journal Information:
Journal Volume: 380; Journal Issue: 22-23; Journal ID: ISSN 0375-9601
Research Org:
Los Alamos National Lab. (LANL), Los Alamos, NM (United States)
Sponsoring Org:
USDOE National Nuclear Security Administration (NNSA)
Contributing Orgs:
Harvard Univ., Cambridge, MA (United States)
Country of Publication:
United States
71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; 97 MATHEMATICS AND COMPUTING; Benford/Reciprocal distribution; Information/Communication theory; Iteration theory