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

• We apply the Law of Total Probability to the construction of scale-invariant pdf's, and require that probability measures be dimensionless and unitless under a continuous change of scales.
• Iterating this procedure for an arbitrary set of normalized pdf's again produces scale-invariant distributions.
• The invariant function of this iteration is given uniquely by the reciprocal distribution, suggesting a kind of universality.
• Requiring maximum entropy for uniformly binned size-class distributions also leads uniquely to the reciprocal distribution.
• We discuss some applications of the above to computation and to the evolution of genomes.

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. We separately demonstrate that the reciprocal distribution results uniquely from requiring maximum entropy for size-class distributions with uniform bin sizes.