Fundamental Flaws in Feller's Classical Derivation of Benford's Law

摘要

Feller's classic text 'An Introduction to Probability Theory and itsApplications' contains a derivation of the well known significant-digit lawcalled Benford's law. More specifically, Feller gives a sufficient condition("large spread") for a random variable $X$ to be approximately Benforddistributed, that is, for $\log_{10}X$ to be approximately uniformlydistributed modulo one. This note shows that the large-spread derivation, whichcontinues to be widely cited and used, contains serious basic errors. Concreteexamples and a new inequality clearly demonstrate that large spread (or largespread on a logarithmic scale) does not imply that a random variable isapproximately Benford distributed, for any reasonable definition of "spread" ormeasure of dispersion