General Distributions of Number Representation Elements

TL;DR

This paper derives general formulas for the joint distributions of significant digits and continued fraction coefficients of arbitrary continuous random variables, revealing deep connections and applications to Benford and Pareto distributions.

Contribution

It introduces unified expressions for digit and continued fraction distributions, generalizes convergence laws, and highlights Pareto variables' broader relevance beyond Benford's law.

Findings

General convergence law of the j-th significant digit distribution

Benford variables dominate asymptotic behavior

Pareto variables extend applicability to scale-invariant phenomena

Abstract

We provide general expressions for the joint distributions of the $k$ most significant $b$-ary digits and of the $k$ leading continued fraction coefficients of outcomes of an arbitrary continuous random variable. Our analysis highlights the connections between the two problems. In particular, we give the general convergence law of the distribution of the $j$-th significant digit, which is the counterpart of the general convergence law of the distribution of the $j$-th continued fraction coefficient (Gauss-Kuz'min law). We also particularise our general results for Benford and Pareto random variables. The former particularisation allows us to show the central role played by Benford variables in the asymptotics of the general expressions, among other results. The particularisation for Pareto variables -- which include Benford variables as a special case -- is specially relevant in the context of pervasive scale-invariant phenomena, where Pareto variables occur much more frequently than Benford variables. This suggests that the Pareto expressions that we produce have wider applicability than their Benford counterparts in modelling most significant digits and leading continued fraction coefficients of real data. Our results may find practical application in all areas where Benford's law has been previously used.